A Note on Order and Index Reduction for Descriptor Systems
aa r X i v : . [ ee ss . S Y ] J a n A Note on Order and Index Reduction forDescriptor Systems
Martin J. Corless and Robert N. Shorten
Abstract —We present order reduction results for linear timeinvariant descriptor systems. Results are given for both forcedand unforced systems as well methods for constructing thereduced order systems. Our results establish a precise connectionbetween classical and new results on this topic, and lead toan elementary construction of quasi-Weierstrass forms for adescriptor system. Examples are given to illustrate the usefulnessof our results.
Index Terms —Descriptor systems, system order reduction,quasi-Weierstrass form
I. I
NTRODUCTION
Descriptor systems have been widely studied in themathematics and engineering literature for several decades[2], [3], [4]. Recently, they have also become very popularin the mainstream control engineering literature, especiallyin the context of switching and hybrid dynamical systems[5], [6], [7], [8], [9], [10], [11], motivated in part, by thefact that descriptor systems provide a natural framework tomodel and analyse many dynamic systems with algebraicconstraints (for example, a mechanical system with coordinateconstraints) [13]. Formally, a descriptor characterization ofa dynamical system consists of a combination of differentialequations and algebraic equations, that coupled togetherdescribe the dynamics of the system under study. Eventhough this formalisation is convenient for many physicaland man-made dynamic systems, the analysis of suchsystems requires bespoke techniques when compared withconventional systems. Our interest in this paper concernslinear time invariant descriptor systems, and methods forcharacterising the qualitative properties of these systems interms of lower order systems. As a special case we alsoconsider reduction methods that yield a standard system; thatis, a system described only by standard differential equationsand no algebraic equations. Our motivation is deriving thesetools is that reduced order characterisations are often usefulthan the corresponding original descriptor characterisationsdue to their compatibility with the broad portfolio of existingresults in Systems Theory which characterise the propertiesof ordinary differential equations. This work builds on ourprevious works on the topic. Order reduction ideas based onfull rank decompositions were first introduced in [15] and[16]. These results were developed further in [12] and [13].The present paper extends our prior work fundamentally ina number of ways. In the original work, one could (in one
M. J. Corless is with the School of Aeronautics and Astronautics, PurdueUniversity, West Lafayette, IN 47906 USA (e-mail: [email protected]). R.N. Shorten is with the Dyson School of Engineering Design, Imperial CollegeLondon, UK (email: [email protected]). Corresponding Author: M. J.Corless reduction step) only reduce a system to one whose index wasone less than the index of the original system; here one canreduce all the way to an index zero system (standard system)in one step. Second, systems with inputs are considered.Third, missing links to established and classical descriptorresults are established, revealing the utility of the approachadvocated here. Finally, new reduced order forms are alsointroduced that are not considered in these previous papers.Specifically our contributions may be summarized as follows.(a) We consider first systems with no input. It is knownthat, subject to some constraints, such a system canbe equivalently represented by a lower order standardsystem. Since the order of a standard system cannot bereduced, this is the lowest order that can be achievedfor the original descriptor system. There are situationswhere it is advantageous to obtain an equivalent systemdescription of lower order but not necessarily of mini-mal lower order.
This occurs, for example in analyzingswitching linear descriptor systems [12], [13]. Our firstset of results is to demonstrate how one can readily obtainvarious equivalent system descriptions of lower order fora linear descriptor system.(b) We also give a simple procedure to reduce a descriptorsystem to an equivalent standard system.Note that, although there are many results in the literature forreducing a descriptor system to a standard system (see [1],for one of the earliest results) there are very few results onreducing to a lower order descriptor system, with the notableexception of [12], and the results therein reduce the index ofthe system by one. The results in this present paper allow oneto reduce a descriptor system to a lower order system of anylower index.(c) In the second part of our paper we consider systemswith inputs and obtain two coupled reduced order sys-tems associated with the original system in descriptorform. These two systems lead directly to the celebratedquasi-Weierstrass form [14] of the original system, butin an elementary manner when compared with existingliterature. Recall the quasi-Weierstrass form gives rise aform that consists of two subsystems which together areequivalent to the original system. One of these subsystemsis a standard system whereas the other is very specialtype of descriptor system called a pure descriptor system.
As stated our derivation provides a simple way of con-structing a quasi-Weierstrass form for a linear descriptorsystem, and relates our approach to existing mathematicalresults on Descriptor systems.ur paper is structured as follows. We present preliminarymaterial in Section 2. Our main results are derived in Sections3 and 4. Examples illustrating the utility of our results are alsogiven on Section 4.II. P
REAMBLE - D
ESCRIPTOR SYSTEMS
Consider a linear time invariant (LTI) system described by thedifferential algebraic equation (DAE) E ˙ x = Ax (1)where x ( t ) ∈ C n is the system state at time t ∈ R and E , A ∈ C n × n . When E is nonsingular, this system is alsodescribed by the standard system ˙ x = E − Ax . If E is singular,then both algebraic equations and differential equationsdescribe the behavior of the system, and the system is knownas a descriptor system. We say that system (1) or ( E , A ) is regular if the polynomialdet ( sE − A ) is nonzero, that is, there exists λ ∈ C such that λ E − A is nonsingular. For such a scalar λ , we can rewritesystem (1) as E ˙ x = ( A − λ E ) x + λ Ex and pre-multiply by ( A − λ E ) − to obtain F ˙ x = ( I + λ F ) x (2)where F : = ( A − λ E ) − E (3)We will find this system description useful for severalpurposes, in particular for reducing system (1) to a system oflower order, that is, lower state dimension.The consistency space C = C ( E , A ) for system (1) or ( E , A ) is the set of all initial states x ∈ C n for which equation (1) hasa classical (that is, differentiable) solution x ( · ) : [ , ∞ ) → C n with the initial condition x ( ) = x . We can characterize thiswith the following concept. The index of a matrix F ∈ C n × n isthe smallest nonnegative integer k ∗ for which rank ( F k ∗ + ) = rank ( F k ∗ ) where rank denotes the rank of a matrix; this indexis zero for a nonsingular matrix. Note that the index of F isalso the smallest nonnegative integer k ∗ for which R ( F k ∗ + ) = R ( F k ∗ ) where R denotes the image or range of a matrix.Also R ( F k ) = R ( F k ∗ ) for all k ≥ k ∗ and R ( F k ) ⊃ R ( F k ∗ ) for k ≤ k ∗ . If F k ∗ = F is nilpotent . Remark 1
It can readily be shown that, for any k = , , , . . . ,the subspace R ( F k ) is the same for all λ for which λ E − A isnonsingular [2]; hence the index of F is the same for all λ forwhich λ E − A is nonsingular; we call this the index of system(1) or ( E , A ) . It is also shown in [2] that C ( E , A ) = R ( F k ) for k ≥ k ∗ where k ∗ is the index of F and for all λ for which λ E − A is nonsingular. Remark 2
Since R ( F k ∗ + ) = R ( F k ∗ ) = C we see that F C = C . This implies that F is a one-to-one mapping of C ontoitself; hence the kernel of F and C intersect only at zero.Note that C = { } if and only if F is nilpotent; in this casewe say that the system is a pure descriptor system and the only differentiable solution is the zero solution x ( t ) ≡
0. If C = { } , we let G be the inverse of the map F restrictedto C , that is, GFx = x and FGx = x when x ∈ C . When thesolution x ( t ) is in C for all t then so is ˙ x ( t ) ; hence multiplying(2) by G results in ˙ x = ˆ Ax (4)where ˆ A = G + λ I . Also multiplying (4) by F results in (2).Thus (4) is equivalent to (2); hence (4) and (1) are equivalent.Thus the restriction of the descriptor system to its consistencyspace is equivalent to the standard system (4) where x ( t ) is in C . III. R EDUCING A DESCRIPTOR SYSTEM
Our first main result, Lemma 3, shows how to simply reducesystem (1) to an equivalent system of lower order and lowerindex. It requires the following concepts and lemmas. For afull column rank matrix X , the matrix X † denotes any left-inverse of X , that is, it satisfies X † X = I where I is an identity matrix. For example, X † = ( X ′ X ) − X ′ .We need the following result for an arbitrary n × n matrix F . Lemma 1:
Suppose F ∈ C n × n , F k = k ≥ X is a matrix of full column rank whose range equals thatof F k . Then, for any integer l ≥ F l X = X ˜ F l where ˜ F = X † FX (5) Proof.
Clearly it holds for l =
0. We now prove, by inductionthat is holds for any l ≥
1. We first show that (5) holds for l =
1, that is, FX = X ˜ F . By assumption, R ( X ) = R ( F k ) ; thus R ( FX ) = R ( F k + ) ⊂ R ( F k ) = R ( X ) that is R ( FX ) ⊂ R ( X ) . So FX = X ˜ F for some matrix ˜ F .Multiplying both sides of this equation by any left-inverse X † of X yields ˜ F = X † FX . Now suppose that (5) holds for someinteger l ≥
1. Then F l + X = FF l X = FX ˜ F l = X ˜ F ˜ F l = X ˜ F l + Thus, (5) holds with l replaced with l +
1. By induction, itholds for all l ≥
1. QEDThe following decomposition is useful in some of the resultsof this paper. Consider any non-zero matrix M ∈ C n × n . A pairof matrices ( X , Y ) is a full rank decomposition of M if X and Y have maximum column rank and M = XY ′ (6)If r is the rank of M then r ≤ n and X , Y ∈ C n × r . Clearly, X and M have the same range while Y and M ′ have the samerange. Also, X = MY † ′ and Y = M ′ X † ′ (7) Lemma 2:
Suppose F ∈ C n × n , F k = k ≥ X , Y is a full rank column rank decomposition of F k . Then, X † FX = Y ′ FY † ′ = : ˜ F (8)nd for any integer l ≥ F l + k = X ˜ F l Y ′ (9) Proof.
Since ( X , Y ) is a full rank column rank decompositionof F k , F k = XY ′ (10)where X , Y are full column rank matrices and the range of X equals that of F k . Thus X = F k Y † ′ and X † FX = X † FF k Y † ′ = X † F k FY † ′ = X † XY ′ FY † ′ = Y ′ FY † ′ Consider any integer l ≥
0. According to Lemma 1, X ˜ F l = F l X ; hence ˜ F l = X † F l X (11)Post-multiplying both sides of (11) by Y ′ and using (10):˜ F l Y ′ = X † F l XY ′ = X † F l F k = X † F l + k (12)Since R ( F l + k ) ⊂ R ( F k ) = R ( X ) , there exists a matrix Y l suchthat F l + k = XY ′ l (13)hence X † F l + k = X † XY ′ l = Y ′ l . It now follows from (12) that Y ′ l = ˜ F l Y ′ Combining this with (13) yields the desired result, F l + k = X ˜ F l Y ′ . QEDWe now obtain our first reduction result. Lemma 3:
Consider a regular descriptor system described by(1) and any λ ∈ C for which λ E − A nonsingular. For anyinteger k ≥ F k =
0, where F is given by (3), let X beany matrix of full column rank whose range equals that of F k .Then, x ( · ) is a differentiable solution to (1) if and only if x = Xz (14)and z ( · ) is a differentiable solution to˜ F ˙ z = ( I + λ ˜ F ) z (15)where ˜ F : = X † FX (16)Moreover z = X † x and the index of (15) is max { k ∗ − k , } where k ∗ is the index of (1) Proof.
When x ( · ) is a differentiable solution to (1) we have x ( t ) ∈ C where C is the consistency space of ( E , A ) . Since C ⊂ R ( F k ) it follows that C ⊂ R ( X ) . Hence, x = Xz and z is uniquely given by z = X † x . As shown earlier, x ( · ) is adifferentiable solution to (1) if and only if it a solution of (2)which is equivalent to FX ˙ z = ( I + λ F ) Xz (17)It follows from Lemma 1 that FX = X ˜ F where ˜ F is given by(16). Thus (15) is equivalent to X ˜ F ˙ z = X ( I + λ ˜ F ) z (18)Since X has maximum column rank, (18) is equivalent to (15).To obtain the index of (15), choose any matrix Y such that ( X , Y ) is a full rank decomposition of F k . Recall from Lemma 2 that for any l ≥ F l + k = X ˜ F l Y ′ . Since X has maximumcolumn rank the matrices F l + k and ˜ F l Y ′ have the same rank.Since Y ′ has maximum row rank the matrices ˜ F l Y ′ and ˜ F l have the same range; hence F l + k and ˜ F l have the same rank.It now follows that if k ≤ k ∗ then the index l ∗ of (15) is k ∗ − k and if k > k ∗ we have l ∗ =
0. QED
Remark 3
For a descriptor system with singular E , the rankof the matrix F is less than n ; thus the rank of F k and, hence, X is less than n . Since X has maximum column rank this tellsus that the state z of the new system in (17) is in C m with m < n . Hence (15) is an equivalent reduced order version ofthe original system (1). Example 1:
To illustrate Lemma 3, consider a descriptorsystem described by (1) with E = − −
22 2 −
20 0 0 , A = − − Since A is non-singular, we can consider λ = F = A − E = − − − − The rank of F is two whereas that of F = (19)and F is one. Thus this is an index two system whoseconsistency space is the range of F . Considering k =
1, thefull column rank matrix X = − − has the same range as that of F . Hence this system can bedescribed by x = Xz and ˜ F ˙ z = z where z = X † x and˜ F = X † FX = (cid:18) − (cid:19) which is an index one matrix. Considering k =
2, the range offull column rank matrix X = (20)is the same as that of F and is the consistency space. Here˜ F = X † FX = −
2. Hence the original descriptor system can bedescribed by the standard system − z = z and x = Xz = [ z z ] T . Also z = X † x = ( x + x ) / n × n matrix F . This shall be used to obtain another reductionresult; namely, Lemma 5. emma 4: Suppose that F ∈ C n × n has index k ∗ and Y isa matrix whose range is the same as that of F ′ k for someinteger k ≥
1. Then, Y ′ F l x = x ∈ R ( F k ∗ ) and all nonnegative integers l . Proof.
Consider any nonnegative integer l . Suppose that Y ′ F l x = x ∈ R ( F k ∗ ) . Since the range of Y isthe same as that of F ′ k , F ′ k = Y ˆ X ′ for some matrix ˆ X and F k = ˆ XY ′ . Hence,0 = ˆ XY ′ F l x = F k F l x = F k + l x (21)Since F has index k ∗ , R ( F k ∗ ) = R ( F k + l + k ∗ ) = F k + l R ( F k ∗ ) thus, F k + l R ( F k ∗ ) = R ( F k ∗ ) . This implies that F k + l is aone-to-one mapping of R ( F k ∗ ) onto itself; hence the kernelof F k + l and R ( F k ∗ ) intersect only at zero. Now (21) impliesthat that x = Lemma 5:
Consider a regular descriptor system described by(1) and any λ ∈ C for which λ E − A nonsingular. For anyinteger k ≥ F k =
0, where F is given by (3), let Y beany matrix of maximum column rank whose range is the sameas that of F ′ k . Then, there is a matrix H such that x ( · ) is adifferentiable solution to (1) if and only if x = Hz (22)and z ( · ) is a differentiable solution to˜ F ˙ z = ( I + λ ˜ F ) z (23)where ˜ F = Y ′ FY † ′ (24)Moreover z = Y ′ x (25)and the index of (23) is max { k ∗ − k , } where k ∗ is the indexof (1). Proof.
As shown earlier, x ( · ) is a differentiable solution to (1)if and only if it is a solution of (2). Introducing ˆ x = F k x weobtain that F ˙ˆ x = ( I + λ F ) ˆ x (26)Using Lemma 3, ˆ x ( · ) is a differentiable solution to (26) if andonly if ˆ x = Xz (27)and z ( · ) is a differentiable solution to˜ F ˙ z = ( I + λ ˜ F ) z (28)where ˜ F = X † FX = Y ′ FY † ′ The second equality comes from Lemma 2. The index of (28)is max { k ∗ − k , } where k ∗ is the index of (1) and z = X † ˆ x = X † F k x = X † XY ′ x = Y ′ x Lemma 4 tells us that the kernel of Y ′ and C intersect onlyat zero, there is a unique matrix H such that (22) holds. QED Example 2:
To illustrate Lemma 5, recall the system inExample 1. We see that Y = − is a full column rank matrix whose range is the same as thatof F ′ . Hence this system can be described by ˜ F ˙ z = z where z = Y ′ x and ˜ F = Y ′ FY † ′ = (cid:18) − (cid:19) which is a index one matrix. Since z = x and x must be inthe range of the matrix X in (20) (the consistency space), wemust have x = [ z z ] T . Considering k = Y = has the same range as that of F ′ . Here ˜ F = Y ′ FY † ′ = − − z = z and z = Y ′ x = x . Since x = z and x must be in the range of the matrix X in (20), we must have x = [ z z ] T . Remark 4
Suppose that ( X , Y ) is a full rank decompositionof the matrix F in (3). Then F = XY ′ . Considering the resultin Lemma 3 for k =
1, we see that the matrix ˜ F in (16) isgiven by ˜ F = X † FX = X † XY ′ X = Y ′ X This along with Lemma 3 and and λ = A is nonsingular. Application to switching linear systems
The above results can be useful in reducing a switching de-scriptor system to a lower order system. To illustrate, considera switching descriptor system described by E σ ( t ) ˙ x = A σ ( t ) x (29)where σ ( t ) ∈ { , , . . . , N } and E i , A i ∈ C n × n for i = , , . . . , N .Suppose that for some λ ∈ C and for each i there exists k i such that the range of F k i i is the same for all i where F i =( A i − λ E i ) − E i . Recalling Lemma 3, let X be any matrix ofmaximum column rank whose range is the same as that of F k i i for all i . Then, x ( · ) is a differentiable solution to (29) if andonly if x = Xz and z ( · ) is a differentiable solution to the lowerorder switching system˜ F σ ( t ) ˙ z = ( I + λ ˜ F σ ( t ) ) z (30)where ˜ F i : = X † F i X . Moreover z = X † x .V. E QUIVALENT STANDARD SYSTEMS
We have already seen that (1) is equivalent to a standardsystem on the consistency space. Here we provide simplecharacterizations of reduced order standard systems whichare equivalent to (1). Lemma 3 leads to the following resultwhich yields an equivalent lower order standard system forthe original descriptor system (1).
Corollary 1:
Consider a regular non-pure descriptor systemdescribed by (1) and any λ ∈ C for which A − λ E nonsingular.With F given by (3) let X be any full column rank matrixwhose range is the same as that of F k for some integer k ≥ k ∗ where k ∗ is the index of ( E , A ) . Then X † FX is nonsingularand x ( · ) is a differentiable solution to (1) if and only if x = Xz (31)and z ( · ) is a differentiable solution to˙ z = ˜ Az (32)where ˜ A = ( X † FX ) − + λ I (33)Moreover z = X † x (34)When A is invertible, one can choose λ =
0. In this case, weobtain the following simpler expressions: F = A − E , ˜ A = ( X † A − EX ) − (35)Lemma 5 leads to the following result which yields anotherequivalent lower order standard system for the originaldescriptor system (1). Corollary 2:
Consider a regular non-pure descriptor systemdescribed by (1) and any λ ∈ C for which λ E − A nonsingular.With F given by (3), let Y be any matrix of maximum columnrank whose range is the same as that of F ′ k for some integer k ≥ k ∗ where k ∗ is the index of ( E , A ) . Then Y ′ FY † ′ isnonsingular and x ( · ) is a differentiable solution to (1) if andonly if x = Y † ′ z (36)and z ( · ) is a differentiable solution to˙ z = ˜ Az (37)where ˜ A = ( Y ′ FY † ′ ) − + λ I (38)Moreover z = Y ′ x (39) Proof.
We just need to show that H = Y † ′ . Since Y ′ x = x in the consistency space C of (1), it follows that { z : z = Y ′ x and x ∈ C } = C m where m equals the dimensionof C and the number of columns of Y . Using (25) and (22)we now obtain that z = Y ′ Hz for all z ∈ C m . Hence Y ′ H = I from which it follows that H = Y † ′ . QED When A is invertible, one can choose λ =
0. In this case, wehave the simpler expressions: F = A − E , ˜ A = ( Y ′ A − EY † ′ ) − (40)The following result leads to further expressions for ˜ A . Lemma 6:
Suppose that F ∈ C n × n is a matrix which isnot nilpotent, has index k ∗ and X and Y are full columnrank matrices whose ranges are the same as that of F k and F ′ k , respectively, for some integer k ≥ k ∗ . Then, Y ′ F l X isnonsingular for every nonnegative integer l . Proof.
Consider any nonnegative integer l and suppose that Y ′ F l Xz =
0. Since the vector Xz is in R ( F k ) and k ≥ k ∗ , thisvector is in R ( F k ∗ ) . It now follows from Lemma 4 that Xz = X has maximum column rank we obtain that z is zero.With Y and X having the same dimensions, Y ′ F l X is square.Thus Y ′ F l X is nonsingular. QED Remark 5
Consider a non-pure system described by (1).Then F k = k where F is givenby (3). Suppose that X and Y are full column rank matriceswhose ranges are the same as that of F k and F ′ k , respectively,where k ≥ k ∗ and k ∗ is the index of F . Then, the above resulttells us that Y ′ X is invertible. Since ( Y ′ X ) − Y ′ X = I , a left-inverse of X is given by X † = ( Y ′ X ) − Y ′ (41)Hence X † FX = ( Y ′ X ) − Y ′ FX (42)and the matrix in (32) is given by˜ A = ( Y ′ FX ) − Y ′ X + λ I (43)Since, ( Y ′ X ) − ′ X ′ Y = I , a left-inverse of Y is given by Y † = ( Y ′ X ) − ′ X ′ (44)Hence Y ′ FY † ′ = Y ′ FX ( Y ′ X ) − and the matrix in (37) is givenby ˜ A = Y ′ X ( Y ′ FX ) − + λ I (45) An equivalent full order standard system on the consistencyspace:
Using the results in Corollary 1 or Corollary 2 we canobtain a standard system which is equivalent to the originaldescriptor system and has the same state as the original system.
Lemma 7:
Consider a non-pure system described by (1).Suppose that X and Y are full column rank matrices whoseranges are the same as that of F k and F ′ k , respectively, where k ≥ k ∗ and k ∗ is the index of F . Then, Y ′ X and Y ′ FX arenonsingular and x ( · ) is a differentiable solution to (1) if andonly if x ( t ) is in the range of X and˙ x = ˆ Ax (46)where ˆ A = X ( Y ′ FX ) − Y ′ + λ X ( Y ′ X ) − Y ′ (47) Proof.
Lemma 6 tells us that Y ′ X and Y ′ FX are nonsingular.It follows from (31), (32) and (34) that the behavior of x isescribed by (46) with ˆ A = X ˜ AX † . Recalling (43) and (42) wesee thatˆ A = X ( Y ′ FX ) − ( Y ′ X )( Y ′ X ) − Y ′ + λ X ( Y ′ X ) − Y ′ = X ( Y ′ FX ) − Y ′ + λ X ( Y ′ X ) − Y ′ One obtains the same result using (36), (37) and (39) alongwith (44) and (45). QEDWhen E is invertible, consider any λ for which A − λ E isinvertible. In this case the index k ∗ of F = ( A − λ E ) − E iszero. Hence X and Y are invertible one can readily show thatˆ A = E − A . When A is invertible, one can choose λ =
0. Inthis case, F = A − E andˆ A = X ( Y ′ A − EX ) − Y ′ (48)V. S YSTEMS WITH INPUTS
We now consider systems with inputs described by E ˙ x = Ax + Bu (49)where u ( t ) ∈ C m is the system input and B ∈ C n × m . When u =
0, a classical solution to (49) is constrained to theconsistency space associated with (49). When u = ( E , A ) is regular, there exists λ ∈ C such that A − λ E isnonsingular and, following the derivation of (2), we see that(49) is equivalent to F ˙ x = ( I + λ F ) x + Gu (50)where F is given by (3) and G : = ( A − λ E ) − B (51)Using the following corollary to Lemma 1 we can obtain ourfirst result, Lemma 8. Corollary 3:
Suppose F ∈ C n × n , F k = k ≥ Y is a matrix of full column rank whose range equals thatof F ′ k . Then, for any integer l ≥ Y ′ F l = ˜ F l Y ′ where ˜ F = Y ′ FY † ′ (52)for l = , , . . . where ˜ F = Y ′ FY † ′ . Lemma 8:
Consider a regular descriptor system described by(49) and any λ ∈ C for which λ E − A nonsingular. For anyinteger k ≥ F k =
0, where F is given by (3), let Y beany matrix of maximum column rank whose range is the sameas that of F ′ k . Suppose x ( · ) is any differentiable solution to(49) and let z = Y ′ x (53)Then z ( · ) is a differentiable solution to˜ F ˙ z = ( I + λ ˜ F ) z + ˜ G u (54)where ˜ F = Y ′ FY † ′ , ˜ G = Y ′ G (55) Proof.
As shown above, x ( · ) is a differentiable solution to (49)if and only if it a solution to (50). Hence Y ′ F ˙ x = Y ′ ( I + λ F ) x + Y ′ Gu Corollary 3 tells us that Y ′ F = ˜ FY ′ where ˜ F = Y ′ FY † ′ ; hence˜ F ˙ z = ( I + λ ˜ F ) z + ˜ G u where z = Y ′ x . QED Remark 6 If k ≥ k ∗ in the above lemma, where k ∗ is the indexof ( E , A ) then, Y ′ FY † ′ is nonsingular; hence (54) is equivalentto the standard system ˙ z = ˜ Az + ˜ B u (56)where˜ A = ( Y ′ FY † ′ ) − + λ I , ˜ B = ( Y ′ FY † ′ ) − Y ′ G (57)With a nonzero input u , the state x is not confined to theconsistency space and we cannot recover x from z . So, nowwe proceed to obtain another reduced order system whichcontains further information on x . To achieve this, need thefollowing result for an arbitrary square matrix F ; this resultis analagous to Lemma 1. Lemma 9:
Suppose that F ∈ C n × n is singular and V is anymatrix of maximum column rank whose range equals thekernel of F k for some integer k ≥
1. Then, for any integer l ≥ F l V = V N l (58)where N = V † FV . Moreover N k = Proof.
We prove this by induction. We first show that (58)holds for l =
1, that is, FV = V N . If v is the range of V , then F k v =
0. Thus F k ( Fv ) = F ( F k v ) =
0; this implies that Fv isin the kernel of F k and, hence, it is in the range of V . Thus R ( FV ) ⊂ R ( V ) . This means that FV = V N for some matrix N . Multiplying both sides of this equation by V † results in N = V † FV . Thus, (58) holds for l =
1. Now suppose that forsome integer l ∗ ≥
1, (58) holds with l = l ∗ . Then F l ∗ + V = FF l ∗ V = FV N l ∗ = V NN l ∗ = V N l ∗ + Thus (58) holds with l = l ∗ +
1. By induction, it holdsfor all l ≥
1. It follows from (58) that F k V = V N k ; hence N k = V † F k V . Since the range of V is the kernel of F k , F k V =
0; thus N k =
0. QEDThe following result is a simple corollary to Lemma 9.
Corollary 4:
Suppose that F ∈ C n × n is singular and W isany matrix of maximum column rank whose range equals thekernel of F ′ k for some k ≥
1. Then, for any integer l ≥ W ′ F l = N l W ′ (59)where N = W ′ FW † ′ . Moreover N k = emma 10: Consider a regular descriptor system described by(49) and any λ ∈ C for which λ E − A nonsingular. For anyinteger k ≥ W be any matrix of maximum column rankwhose range is the same as that of the kernel of F ′ k with F given by (3). Suppose x ( · ) is a differentiable solution to (49)and let z = W ′ x (60)Then z ( · ) is a differentiable solution to˜ N ˙ z = z + ˜ B u (61)where˜ N = ( I + λ W ′ FW † ′ ) − W ′ FW † ′ , ˜ B = ( I + λ W ′ FW † ′ ) − W ′ G (62)and ˜ N k = Proof.
As shown earlier, x ( · ) is a differentiable solution to (49)if and only if it a solution to (50). Hence W ′ F ˙ x = W ′ ( I + λ F ) x + W ′ Gu From Corollary 4, W ′ F = NW ′ where N = W ′ FW † ′ and N k =
0; hence N ˙ z = ( I + λ N ) z + W ′ Gu where z = W ′ x . Since N k = N are zero;hence the eigenvalues of I + λ N are one, so I + λ N is invertibleand we obtain the desired result that ( I + λ N ) − N ˙ z = z + ( I + λ N ) − W ′ Gu Since ( I + λ N ) − and N commute, ˜ N = ( I + λ N ) − N and N k =
0, it follows that ˜ N k = ( I + λ N ) − k N k =
0. QED
A. Quasi-Weierstrass form
We have obtained two subsystems (54) and (61) associatedwith the original descriptor system (49). In order for thesetwo subsystems to completly describe the behavior of theoriginal system, we need the matrix [ Y W ] to be nonsingular.This turns out to be the case if we consider k ≥ k ∗ , theindex of the original system. To prove this we first obtain thefollowing result for an arbitrary square matrix. Lemma 11:
Suppose F ∈ C n × n is singular, is not nilpotent,has index k ∗ and V , W are any matrices of maximum columnrank whose ranges are the kernels of F k and F ′ k , respectively,for some k ≥ k ∗ . Then V ′ W is nonsingular. Proof.
To show that V ′ W is nonsingular, suppose V ′ W z = W z is in the orthogonal complement of the range of V which equals the range of F ′ k . Hence W z = Y ξ for some vector ξ where Y is a full column rank matrix whose range equalsthat of F ′ k . Let X be a full column rank matrix whose rangeequals that of F k . Then the range of X equals the orthogonalcomplement of the range of W and X ′ Y ξ = X ′ W z =
0. Lemma6 tells us that X ′ Y = ( Y ′ X ) ′ is nonsingular. Thus ξ is zero andsince W has maximum column rank, z =
0. This implies that V ′ W is nonsingular. QED We can now prove that T = [ Y W ] is invertible for k ≥ k ∗ . Lemma 12:
Suppose F ∈ C n × n is singular, is not nilpotent andhas index k ∗ . For any k ≥ k ∗ , let X and Y be any matrices ofmaximum column rank whose ranges are the same as that of F k and F ′ k , respectively, and let V and W be any matrices ofmaximum column rank whose ranges are the kernels of F k and F ′ k , respectively, Then [ Y W ] is nonsingular with inverse (cid:20) ( X ′ Y ) − X ′ ( V ′ W ) − V ′ (cid:21) (63) Proof.
Since k ≥ k ∗ , where k ∗ is the index of F , we know fromLemma 6 and Lemma 11 that X ′ Y and V ′ W are nonsingular.Since the range of W is the kernel of F ′ k we have F ′ k W = W ′ F k =
0. Since the range of X is F k , we must have X ′ W = ( W ′ X ) ′ =
0. Using the same reasoning we also have Y ′ V =
0. Hence (cid:20) ( X ′ Y ) − X ′ ( V ′ W ) − V ′ (cid:21) (cid:2) Y W (cid:3) = (cid:20) ( X ′ Y ) − X ′ Y ( X ′ Y ) − X ′ W ( V ′ W ) − V ′ Y ( V ′ W ) − V ′ W (cid:21) = (cid:20) I I (cid:21) QEDUsing the above lemma along with Remark 6 and Lemma10 we obtain a decomposition of the original system intoa standard system and a pure descriptor system. Thisdecomposition is obtained in [14] and is referred to as a quasi-Weierstrass form of (49). The derivation in [14] isbased on the Wong sequences presented in [17]. We believethe derivation here is more elementary. Also, one may simplycompute the matrices involved here by performing a singularvalue decomposition of F k where k is greater than or equalto the index of ( E , A ) ; see Remark 7 below. Theorem 1:
Consider a regular non-pure descriptor system ofindex k ∗ described by (49) with E singular and any λ ∈ C for which λ E − A is nonsingular. With F given by (3) and forany integer k ≥ k ∗ , let X and Y be any matrices of maximumcolumn rank whose ranges are the same as that of F k and F ′ k ,respectively, and let V and W be any matrices of maximumcolumn rank whose ranges equal the kernels of F k and F ′ k ,respectively. Then x ( · ) is a differentiable solution to (49) ifand only if x = X ( Y ′ X ) − z + V ( W ′ V ) − z (64)and ˙ z = ˜ Az + ˜ B u (65)˜ N ˙ z = z + ˜ B u (66)where ˜ A and ˜ B are given by (57) while ˜ N and ˜ B are givenby (62). Moreover ˜ N k = z = Y ′ x , z = W ′ xxample 3: To illustrate Theorem 1, consider descriptor sys-tem (49) with A and E as given in Example 1 and B = Here k ∗ = F is given in (19). From this one may readilyobtain X = , Y = , V = , W = − which results in˜ A = − . , ˜ B = , ˜ N = (cid:20) − − (cid:21) , ˜ B = (cid:20) . . (cid:21) and x = z + − z Remark 7
In general, one can reliably obtain the matrices X , Y , V , W from a singular value decomposition of F k where k is greater than or equal to the index of F . Specifically, supposethat F k = (cid:2) U U (cid:3) (cid:20) Σ
00 0 (cid:21) (cid:2) V V (cid:3) ′ is a singular value decomposition of F k where Σ is diagonalwith diagonal elements equal to the nonzero singular valuesof F k , then X = U , Y = V , V = U , W = V (67) Remark 8 (Discrete-time systems)
Clearly the results ofthis paper can be applied to discrete-time descriptor systemsdescribed by the difference algebraic equation Ex ( t + ) = Ax ( t ) (68)where x ( t ) ∈ C n is the system state at time t ∈ N and E , A ∈ C n × n . This is because all the results of this paper areonly concerned with the pair ( E , A ) and to obtain discrete-timeresults just replace ˙ x with x ( t + ) .VI. C ONCLUSIONS
In this paper we have obtained order and index reductionresults for linear time invariant descriptor systems. Resultsare given for both forced and unforced systems as wellmethods for constructing the reduced order systems. Resultsare also derived that relate our results to existing results in theliterature. Future work will consider developing similar resultsfor classes of nonlinear descriptor systems. R
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