Scalable Positivity Preserving Model Reduction Using Linear Energy Functions

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1 Salable Positivity Preserving Model Redution Using Linear Energy Funtions Sootla, Aivar; Rantzer, Anders Published in: IEEE 51st Annual Conferene on Deision and Control (CDC), 2012 DOI: /CDC Published: Link to publiation Citation for published version (APA): Sootla, A., & Rantzer, A. (2012). Salable Positivity Preserving Model Redution Using Linear Energy Funtions. In IEEE 51st Annual Conferene on Deision and Control (CDC), 2012 (pp ). IEEE--Institute of Eletrial and Eletronis Engineers In.. DOI: /CDC General rights Copyright and moral rights for the publiations made aessible in the publi portal are retained by the authors and/or other opyright owners and it is a ondition of aessing publiations that users reognise and abide by the legal requirements assoiated with these rights. Users may download and print one opy of any publiation from the publi portal for the purpose of private study or researh. You may not further distribute the material or use it for any profit-making ativity or ommerial gain You may freely distribute the URL identifying the publiation in the publi portal Take down poliy If you believe that this doument breahes opyright please ontat us providing details, and we will remove aess to the work immediately and investigate your laim. L UNDUNI VERS I TY PO Box L und

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3 51st IEEE Conferene on Deision and Control Deember 10-13, Maui, Hawaii, USA Salable Positivity Preserving Model Redution Using Linear Energy Funtions Aivar Sootla Department of Bioengineering, Imperial College London South Kensington Campus, SW3 2AZ, London, United Kingdom Anders Rantzer Department of Automati Control, Lund University Box 118 SE Lund, Sweden Abstrat In this paper, we explore positivity preserving model redution. The redution is performed by trunating the states of the original system without balaning in the lassial sense. This may result in onservatism, however, this way the physial meaning of the individual states is preserved. The redued order models an be obtained using simple matrix operations or using distributed optimization methods. Therefore, the developed algorithms an be applied to sparse large-sale systems. Index Terms model redution; positive systems; nonnegative matries I. INTRODUCTION Model order redution is one of the lassial problems in systems theory with a vast number of methods reported in the literature (f. [1], [2]). The lassial model redution tehniques are balaned trunation ([3]), optimal Hankel approximation ([4]), methods based on Krylov subspaes ([2], [5]). However, all these methods do not, in general, preserve some properties of the systems. Positivity is one of these properties. Positive systems are linear time-invariant systems, whih will always have a nonnegative state vetor x given a nonnegative initial ondition and nonnegative input signals. Positive systems are met in many appliations, suh as networked systems, graphs, systems biology et. In some appliations the systems are extremely large, whih is a major bottlenek for analysis. Clearly, model redution ould failitate analysis or simulation of suh systems. However, positivity preserving model redution is a relatively new and not extensively studied topi. In a general setting, the authors are aware of only a few published referenes, these are[6],[7]. These methods require solving large linear matrix inequalities and therefore, onsiderable omputational effort to obtain an approximation. We should also mention [8] and [9], whih deal with a similar problem setting. The positive systems possess a number of extraordinary properties. For example, Lyapunov funtions an be linear with respet to the state-spae vetor x, while in general they are quadrati (f. [10]). The omputation of the H norm requires simple matrix operations, in general, it requires solution of matrix Riati equations. The goal of this paper is to extend this list to model redution. Although, the algorithms obtained in this paper are far from optimal in the H norm, they are salable and require simple matrix manipulations. Therefore, the presented algorithms an be used for large-sale systems, while [6], [7] employ linear matrix inequalities and an be applied to systems of relatively low-order. The paper is organized as follows. In Setion II theoretial bakground of model redution is skethed and a theoretial base for the positivity preserving redution proedure is outlined. In Setion III the positivity preserving redution methods are presented and in Setion IV these are illustrated on numerial examples. All the propositions, whih are mostly known fats about nonnegative matries, are formulated and proved in Appendix. Notation Throughout the paper we use standard notation for spaes of sequenes l p, as well as standard H ( H ) and Hankel ( H ) norms (f. [1]). Let R+ n m stand for the positive orthant of R n m. Let also A denote the transpose of a matrix A, and A B stand for an entry-wise inequality for matries of the same size A and B. We say that a matrix is Shur stable, or simply Shur, if all its eigenvalues have absolute values smaller than one. A matrix A is alled nonnegative if A 0 and A 0, and it is alled positive if A > 0. Given a partition of matries ( ) ( ) A11 A A = 12 B1 B = C = ( ) C A 21 A 22 B 1 C 2 2 let the omplements to A 22 be A 1 = A 11 +A 12 (I A 22 ) 1 A 21 B 1 = B 1 +A 12 (I A 22 ) 1 B 2 C 1 = C 1 +C 2 (I A 22 ) 1 A 21 D 1 = D +C 2(I A 22 ) 1 B 2 Similarly, let A 2, B 2, C 2, D 2 be the omplements to A 11. Note that inverses of I A 11 and I A 22 exist if A is a nonnegative Shur matrix. In fat, in this ase A 11 and A 22 are also nonnegative Shur ([10]). II. ENERGY FUNCTIONS, HANKEL NORMS AND MODEL REDUCTION In this paper we fous on disrete-time positive systems, whih means that the state-spae matries A, B and C have only nonnegative entries. However, the same tehniques an be suessfully applied to ontinuous time models (in this /12/$ IEEE 4285

4 ase, all entries of B and C, and only off diagonal entries of A are nonnegative). Let G be a salar-valued positive system: { x(k +1) = Ax(k)+Bu(k) G = (1) y(k) = Cx(k) where A R+ n n, B Rn 1 +, and C R1 n +. Let also A be Shur stable. In lassial model redution algorithms the, so alled, energy funtions are employed: L o (x 0,l 2 ) = y l2 [0,+ ) L (x 0,l 2 ) = min u u l2(,0] where x(0) = x 0, x( ) = 0, u(k) = 0 for all k 0. The funtion L o is alled the observability energy funtion and L the ontrollability one, and they an be readily omputed L o (x 0,l 2 ) = x 0,Qx 0 1/2 L (x 0,l 2 ) = x 0,P 1 x 0 1/2 A QA Q+C C = 0 APA P +BB = 0 These funtions are also used to define the norm of the Hankel operator indued by l 2 signals (or simply the Hankel norm, f. [11]): y l2[0,+ ) L o (x 0,l 2 ) G H = max = max x 0, u u l2 (,0] x 0 L (x 0,l 2 ) Now, for model redution purposes one an use diretly the matries P and Q (see, [3]), whih yields a simple, but powerful heuristi. Or one an ompute an optimal approximation in the Hankel norm ([4]). The funtions L 2 o and L 2 an be used as Lyapunov funtions for the dynamial system (1). It is known that for positive systems as (1) there are linear Lyapunov funtions in the form v x for some nonnegative vetor v (f. [10]). This lead the authors to an assertion of the existene of L o and L with a similar property. Unfortunately this assertion is only partially true. The observability energy funtion with suh a property is easy to obtain, simply by hanging the norm of the output signal: L o (x 0,l 1 ) = y l1[0,+ ) where x(0) = x 0 and u(k) = 0 for all nonnegative k. Sine the model is onfined to a positive orthant, x 0 and CA k are nonnegative, whih entails that: L o (x 0,l 1 ) = y l1[0,+ ) = CA k x 0 Using the well-known matrix series formula (I A) 1 = A k (see, Proposition 1 in Appendix for a simple proof), the expression above an be omputed as: L o (x 0,l 1 ) = q x 0 where q = C(I A) 1 The vetor q an be omputed by inverting the matrix I A or using distributed programming methods as in [12]. Note, if A is Shur stable then the matrix (I A) 1 is nonnegative ([10]). This means that for stable G, the vetor q is nonnegative, as well. It is possible to introdue a dual toq vetor,i.e.,p = (I A) 1 B.However,theontrollability funtions L (x 0,l 1 ) or L (x 0,l ) do not expliitly depend on p, whih ertainly ompliates any redution proedure. However, by introduing a notion similar to the Hankel norm, but indued by l 1 and l signal norms, an easy approximation proedure an be obtained. Introdue: G H,1, = max x 0 R n +, u y l1[0,+ ) u l (,0] where y = Gu, x(0) = x 0, x( ) = 0, u(k) = 0 k 0. Lemma 1: Let G be a positive model with the state x onfined to R n +, then G H,1, = C(I A) 2 B = q p Proof. First, show that G H,1, C(I A) 2 B. Sine x 0 = A m Bu( m), then y l1 = m=0 CA k x 0 = C(I A) 1 A m Bu( m) m=0 Now using the Hölder s inequality we get: C(I A) 1 A m Bu( m) C(I A) 1 A m B l1 u l m=0 It anbeverifiedthat C(I A) 1 A m B l1 = C(I A) 2 B and finally: G H,1, = max x 0, u C(I A) 2 B u l (,0] u l (,0] y l1[0,+ ) u l (,0] = C(I A) 2 B To prove the onverse, let p = (I A) 1 B. If we hoose u( k) equal to one for all k, then the state x 0 is equal to A k B and therefore x 0 = p. This gives us that y l1[0,+ ) = u l (,0] = 1 CA k (I A) 1 B = C(I A) 2 B and G H,,1 C(I A) 2 B. Remark 1: Thenorm H,1, doesnothangebyadding the onstraint u( k) 0, sine the maximizing sequene u( k) is nonnegative as shown above. Remark 2: In the multivariable ase(for the matrix-valued G with m 2 outputs and m 1 inputs) where C = m2 i=1 G H,1, = C(I A) 2 B C i, B = m1 i=1 B i andb i denotestheindividual olumns of B, C i denotes the individual rows of C. Remark 3: It an be verified using similar tehniques that: 1) G H,1,1 = C(I A) 1 B = q (I A)p 2) G H,, = C(I A) 1 B = q (I A)p 3) G H,,1 = max k 0 CAk B 4286

5 Note that G H,1,1 = G H,, = C(I A) 1 B. Reall that for salar-valued G with D = 0 the H norm is also equal to C(I A) 1 B. Moreover, all indued norms are equal (f. [12]). Whih gives us an unexpeted result that for r = 1 and r = : { y lr [0,+ ) G H,r,r = max x(0) = x } 0, = x 0 R n +,u x( ) = 0 u lr (,0] y lr [,+ ) max u u lr (,+ ] = G r ind Note, the relation G H,2,2 G 2 ind does not generally hold, where G H,2,2 is the Hankel norm and G 2 ind is the H norm. This fat beomes lear in light of results in [13]. Based on [13] it an be shown that for salar-valued positive systems the sum of the Hankel singular values of G is equal to G 2 ind /2. At the same time, G H,2,2 is equal to the maximal Hankel singular value of G. The vetorspand q are not lassial Gramians,however,one of their properties an be very useful: Lemma 2: Ifanentryofthevetorp(thevetorq)isequal to zero, then the orresponding state is not ontrollable (not observable). Proof. Without loss of generality, we prove only the ontrollability part. By Proposition 3 (see, Appendix), for an arbitrary partitioning of the vetor p = ( p 1 p 2) we have p 2 = (I A 2 ) 1 B 2 and p 1 = (I A 1 ) 1 B 1. Now let p 2 be a zero vetor and p 1 be a positive vetor, whih leads to: and thus (I A 2) 1 B 2 = 0 0 = B 2 = B 2 +A 21 (I A 11 ) 1 B 1 The sum of nonnegative matries an be zero, only if the summands (i.e., B 2 and A 21 (I A 11 ) 1 B 1 ) are zero matries. Now, we have (I A 22 ) 1 A 21 (I A 11 ) 1 B 1 = 0 Due to Proposition 2 and the fat that B 1 = B 1 (due to B 2 = 0) (I A 22 ) 1 A 21 (I A 11 ) 1 B 1 = 0 Sine (I A 11 ) 1 B 1 is equal to the positive vetor p 1, (I A 22 ) 1 A 21 is a zero matrix, whih also proves that A 21 is a zero matrix. Finally, due to A 21 and B 2 being zero matries, the system G is not ontrollable. Having a zero entry in p is a suffiient ondition for unontrollability. In fat, it is fairly easy to onstrut unontrollable systems with a positive vetor p. But, a more interesting question is how p related to positive ontrollability, whih is not addressed in this paper. To onlude the setion, we remark that vetors p and q are appearing in various norms and have interesting properties. However, it is not entirely lear why and how p and q measure the importane of states in the H norm, but there ertainly is a onnetion. III. MODEL REDUCTION METHODS In relation to p = (I A) 1 B and q = C(I A) 1, introdue the vetor σ as σi 2 = q i p i for all entries i. This vetor determines the weight of eah state in the inputoutput relationship. The first step in the redution proedures presented in this setion is to determine an invertible matrix T, suh that the new state-spae matries are TAT 1, TB, CT 1. Moreover, T(I A) 1 B = (C(I A) 1 T 1 ) = σ and σ i σ j if i j. Suh T is the produt of a permutation and the diagonalmatrix with entries q i /p i on the diagonal, whih also implies that T and its inverse are nonnegative matries. If p and q have zero entries, these states an be trunated, sine they orrespond to unontrollable or unobservable states (based on Lemma 2). Based on σ partition the state-spae as: x = ( x1 x 2 ) A = ( ) ( ) ( ) A11 A 12 B1 C B = C A 21 A 22 B = 1 2 C 2 where x 2 orresponds to small entries of σ. A. Model Redution by Trunation of States The simplest version of model redution in this setting is the state trunation, whih is summarized in Algorithm 1. Basially, we throw away the states orresponding to small σ i, whih determine the weight of everystate in input-output relationship.the H errorof approximationin this ase an be readily omputed. Lemma 3: Assume G is positive and asymptotially stable. Let G tr be a redued order system obtained from Algorithm 1, then G tr is positive and asymptotially stable. Moreover, G G tr H = G(1) G tr (1) = q 2(I A 2)p 2 Proof. It is straightforward to show that G tr is positive and asymptotially stable, therefore, we ontinue with the proof of the error expression. The transfer funtion G G tr an be written using impulse response as G(z) = (CA k B C 1 (A 11 ) k B 1 )z k i=0 Due to Proposition 4 (see, Appendix) we have and therefore (A k ) 11 (A 11 ) k CA k B C 1 (A 11 ) k B 1 whih means that G G tr has a positive impulse response. Using the same proof as in [12] it an be shown, that G G tr H = (CA k B C 1 A k 11 B 1) i=0 and G G tr H = G(1) G tr (1). Due to Proposition 5 (see, Appendix) C(I A) 1 B = C 2(I A 2) 1 B 2+C 1 (I A 11 ) 1 B 1 whih entails that G(1) G tr (1) = C 2(I A 2) 1 B

6 Aording to Proposition 3 (see, Appendix) q 2 = C2(I A 2 ) 1 and p 2 = (I A 2 ) 1 B2, and finally: C 2(I A 2) 1 B 2 = q 2(I A 2)p 2 Algorithm 1 Model Redution Based on Trunation Perform a state-spae transformation suh that σ i = p i = q i and σ i σ j if i j Based on σ partition the state-spae as: ( ) ( ) ( ) ( ) x1 A11 A x = A = 12 B1 B = C x 2 A 21 A 22 B C = 1 2 C 2 where x 2 orresponds to small entries of σ. Compute the redued order model as: [ ] A11 B G tr = 1 C 1 0 B. Model Redution Based on Singular Perturbation Consider the ase of trunation based on singular perturbation as in [14]. Sine for positive models the largest value G(e jω ) ours at ω = 0, it seems more important to math the DC gains of the full and the redued order models. Trunation of the positive systems based on singular perturbation is onluded in Algorithm 2, whih possesses the following properties: Lemma 4: Assume G is positive and asymptotially stable. Let G sp be obtained using Algorithm 2, then: The system G sp is positive and asymptotially stable The H norms of the models math, i.e., G H = G sp H The norm G sp H,1, is equal to q 1 p 1 = G H,1, q 2p 2 Proof. For the proof of stability the reader is referred to [14]. Therein it is shown that G(1) = G sp (1), i.e., the DC gains math. Let us show positivity of the redued order model. Sine A 22 is nonnegative and asymptotially stable, the matrix (I A 22 ) 1 is also nonnegative (f. [10]). This implies that matries A 1, B 1 and C 1 are nonnegative and G sp is a positive system. This ompletes the proof of the seond bullet, sine G sp H = G sp (1) and G sp (1) is equal to G H = G(1). Aording to Proposition 3 (see, Appendix) it is possible to deompose q p into the sum q 1p 1 +q 2p 2, where q 1 p 1 = C 1 (I A 1 ) 2 B 1 Note that A 1, B 1, C 1 is the state-spae representation of G sp. Therefore, G sp H,1, = q 1 p 1 = G H,1, q 2 p 2 Computing the H approximation error for Algorithm 2 is more demanding than for Algorithm 1, sine G G sp does not have a positive impulse response and Riati equations have to be involved. Moreover, the approximation error G G sp H does not expliitly depend on σ, whih seems as a weaker theoretial result. However, numerial experiments show that Algorithm 2 is ompetitive in omparison to Algorithm 1 and many examples the former outperforms the latter. Algorithm 2 Model Redution Based on Singular Perturbation Perform a state-spae transformation suh that σ i = p i = q i and σ i σ j if i j Based on σ partition the state-spae as: ( ) ( ) ( ) ( ) x1 A11 A x = A = 12 B1 B = C x 2 A 21 A 22 B C = 1 2 C 2 where x 2 orresponds to small entries of σ. Compute the redued order model as: [ ] A G sp = 1 B1 C1 D1 IV. EXAMPLES The major interest in the numerial examples is omparing the presented algorithms to [7], where a somewhat similar proedure was employed. In[7] the states are trunated based on diagonal matries P and Q, whih are the solutions to linear matrix inequalities. Let [7]-tr be a simple trunation method from [7], and [7]-sp be a redutionbased on singular perturbation from [7]. For brevity, we are going to use the following notation for the state-spae representations: [ ] A B G = C D Example 1: First, we are going to ompare the Algorithms 1 and 2 to algorithms from [7] on random examples, whih depit learly the overall trends. Consider the positive systems [ ] [ ] A B1 A B2 G 1 = and G C = C A = B 1 = ( ) T C 1 = ( ) B 2 = ( ) T C2 = ( ) Both systems are ontrollable and observable. Tables I and II depit the result of redution proedures. It is notieable that Algorithm 1 performs similarly to [7]-tr, while Algorithm 2 performs similarly to [7]-sp. This is due the struture of trunation, whih is quite similar in both approahes. The only 4288

7 TABLE I APPROXIMATION ERRORS100 G 1 (G 1 ) r H / G 1 H IN EXAMPLE 1 Redution order k Algorithm Algorithm [7]-tr [7]-sp TABLE II APPROXIMATION ERRORS100 G 2 (G 2 ) r H / G 2 H IN EXAMPLE 1 Redution order k Algorithm Algorithm [7]-tr [7]-sp differene is the way of determining the state s ontribution into the input-output relationship. However, in some ases the presented algorithms perform better than their ounterparts from [7], and in some ases it is the opposite. Authors at the present moment annot elaborate on the reasons for suh a behavior. It an be related to properties of matries B and C, however, no lear onnetion has been established so far. The methods from [7] are muh more numerially demanding than the presented algorithms, sine [7] require solving linear matrix inequalities. Taking into aount the performane in terms of the approximation errors, we an state that our approah has a few advantages over [7]. Example 2: The seond example is a ontinuous-time multi-input-multi-output positive system desribed in [6]. Even though expliit H approximation errors were not omputed for the matrix valued ase, we an still ompare the performane of the algorithms. In this example, the Algorithms 1 and 2 have exatly the same results as algorithms from [7]. In [6] the redution order was set to 2, whih resulted in G G r H / G H = 0.04 relative error. Reall, that using Algorithms 1 and 2 we an preserveaninterpretationofthestatesx i,whenwith[6] suh interpretation is lost. Moreover, the iterative semidefinite approah [6] does not seem to be ompetitive for large systems due to numerial effiieny. G = ACKNOWLEDGMENT This work is performed at Department of Automati Control and is sponsored by LCCC Lund Center for Control of Complex Engineering Systems funded by the Swedish Researh Counil TABLE III APPROXIMATION ERRORS G G r H / G H IN EXAMPLE2 Redution order k Algorithm Algorithm V. CONCLUSION This paper presents two model redution algorithms whih preserve stability and positivity. The algorithms are salable and an be applied to sparse large-sale systems, for whih inversion of matrix A an be performed effiiently. As an alternative one an employ distributed methods to ompute the vetors p and q. The main aspet of the algorithm is that balaning in the lassial sense is not performed, while omputing the approximation. This fat an be seen as a drawbak or an advantage depending on the view point. The methods are onservative with respet to balaned trunation, in fat, even first order balaned redued order models sometimes have a better math than all redued order models obtained by the presented algorithms. However, the presented algorithms preserve the interpretation of the individual states of the systems, whih an be a good asset for analysis. REFERENCES [1] G. Obinata and B. Anderson, Model Redution for Control System Design. London,UK: Springer-Verlag, [2] A. C. Antoulas, Approximation of Large-Sale Dynamial Systems (Advanes in Design and Control). Philadelphia, PA, USA: Soiety for Industrial and Applied Mathematis, [3] B. Moore, Prinipal omponent analysis in linear systems: Controllability, observability, and model redution, IEEE Trans. on Automati Control, vol. 26, no. 1, pp , Feb [4] K. Glover, All optimal Hankel-norm approximations of linear multivariable systems and their L -error bounds, International Journal of Control, vol. 39, pp , [5] S. Gugerin, A. C. Antoulas, and C. Beattie, H 2 model redution for large-sale linear dynamial systems, SIAM Journal on Matrix Analysis and Appliations, vol. 30, no. 2, pp , [6] P. Li, J. Lam, Z. Wang, and P. Date, Positivity-preserving model redution for positive systems, Automatia, vol. 47, no. 7, pp , [7] T. Reis and E. Virnik, Positivity preserving model redution, in Positive Systems. Springer Berlin / Heidelberg, 2009, vol. 389, pp [8] T. Ishizaki, K. Kashima, and J. Imura, Extration of 1-dimensional reation-diffusion struture in siso linear dynamial networks, in Pro. of IEEE Conf. on Deision and Control, de. 2010, pp [9] B. Munsky and M. Khammash, The finite state projetion algorithm for the solution of the hemial master equation, The Journal of Chemial Physis, vol. 124, no. 4, p , [10] L. Farina and S. Rinaldi, Positive linear systems: theory and appliations. John Wiley & Sons, [11] K. Zhou, J. C. Doyle, and K. Glover, Robust and Optimal Control. Upper Saddle River, NJ, USA: Prentie-Hall, In., [12] A. Rantzer, Distributed ontrol of positive systems, in 50th IEEE Conferene on Deision and Control and European Control Conferene, Orlando, Florida, USA, De [13] K. Fernando and H. Niholson, On the struture of balaned and other prinipal representations of siso systems, Automati Control, IEEE Transations on, vol. 28, no. 2, pp , feb [14] Y. Liu and B. Anderson, Singular perturbation approximation of balaned systems, in Pro. of IEEE Conf. on Deision and Control, vol. 2, de 1989, pp

8 APPENDIX Proposition 1: Let A be a Shur matrix with A, B and C being nonnegative matries, then p = A k B = (I A) 1 B and q = CA k = C(I A) 1. Proof. We are going to show only that CA k = C(I A) 1, the seond statement an be obtained by transposing the relations. Assume CA k = q, then q A = CA k = CA k C = q C k=1 Finally, q = C(I A) 1. Proposition 2: Let A R n n B R k n, C R n k and D R k k. If A, D, A BD 1 C and D CA 1 B are invertible, then: (A BD 1 C) 1 BD 1 = A 1 B(D CA 1 B) 1 Proof. Assume the assertion is true. (A BD 1 C) 1 BD 1 = A 1 B(D CA 1 B) 1 multiply from the left with A BD 1 C and from the right with D CA 1 B BD 1 (D CA 1 B) = (A BD 1 C)A 1 B B BD 1 CA 1 B = B BD 1 CA 1 B We arrive to identity valid for any matries A, B, C, D satisfying the assumptions of the proposition. This onludes the proof. Proposition 3: Let A be a Shur, nonnegative matrix. Given an arbitrary partitioning of q = ( q 1 q 2 ) q 1 = C 1 (I A 1 ) 1 q 2 = C 2(I A 2) 1 Proof. The proposition is going to be proved by diret omputation. First it is required to ompute the inverse of the matrix I A: ( I A11 A 12 A 21 ) 1 = I A 22 ( ) X11 X 12 X 21 X 22 The entries X 11 and X 12 are omputed in a straightforward manner using the Shur s omplement: X 11 = (I A 11 A 12 (I A 22 ) 1 A 21 ) 1 = (I A 1) 1 X 21 = (I A 22 ) 1 A 21 (I A 1 ) 1 X 12 = (I A 1) 1 A 12 (I A 22 ) 1 Using Proposition 2 rewrite the expression for X 12 as: X 12 = (I A 1 ) 1 A 12 (I A 22 ) 1 = (I A 11 ) 1 A 21 (I A 2) 1 Using the same proposition it is possible to show that: X 22 = (I A 2 ) 1 All is required to show now is that: q 1 = C 1X 11 +C 2 X 21 = C 1 (I A 1 ) 1 q 2 = C 1 X 12 +C 2 X 22 = C 2(I A 2) 1 whih follows immediately given the matrix X. Proposition 4: For any nonnegative matrix A and any positive integer k the following inequality is true: (A k ) 11 (A 11 ) k Proof. A similar statement an be found in [9] for the ontinuous time systems. Let us prove the proposition by indution. For k = 1, the assertion is obviously valid, sine A 11 A 11. Assume (A k ) 11 (A 11 ) k, let us show it for k +1: (A k+1 ) 11 = (A k A) 11 = (( )( )) (A k ) 11 (A k ) 12 A11 A 12 (A k ) 21 (A k ) 22 A 21 A 22 (A k ) 11 A 11 +(A k ) 12 A 21 (A 11 ) k A 11 = (A 11 ) k+1 The last inequality is valid sine matries (A k ) 12 and A 21 are nonnegative and (A k ) 11 (A 11 ) k by assumption. Proposition 5: Let A be a Shur, nonnegative matrix, then the following equality holds: C(I A) 1 B = C 1(I A 1) 1 B 1+C 2 (I A 22 ) 1 B 2 Proof. Using the Shur s omplement it is possible to rewrite (I A) 1 in terms of blok matries A ij as: ( ) 1 ( ) I A11 A 12 I 0 = A 21 I A 22 (1 A 22 ) 1 A 21 I ( )( ) (I A 1 ) 1 0 I A12 (1 A 22 ) 1 0 (1 A 22 ) 1 0 I Note that ( ) ( ) ( ) C I 0 1 C 2 = C1 C 2 (1 A 22 ) 1 A 21 I ( ) ( )( ) B 1 I A12 (1 A = 22 ) 1 B1 B 2 0 I B 2 Therefore C(I A) 1 B = ( C1 C 2 ) ( )( ) (I A 1 ) 1 0 B 1 0 (1 A 22 ) 1 = B 2 11 = C 1 (I A 1 ) 1 B 1 +C 2(I A 22 ) 1 B