Circuits and Systems, 2011, 2, 261268 doi:10.4236/cs.2011.24036 Published Online October 2011 (http://www.scirp.org/journal/cs) Copyright © 2011 SciRes. CS New Stability Tests of Positive Standard and Fractional Linear Systems* Tadeusz Kaczorek Faculty of Electrical Engineering, Bialystok University of Technology, Bialystok, Poland Email: kaczorek@isep.pw.edu.pl Received March 1, 2011; revised June 8, 2011; accepted June 15, 2011 Abstract New tests for checking asymptotic stability of positive 1D continuoustime and discretetime linear systems without and with delays and of positive 2D linear systems described by the general and the Roesser models are proposed. Checking of the asymptotic stability of positive 2D linear systems is reduced to checking of suitable corresponding 1D positive linear systems. It is shown that the stability tests can be also applied to checking the asymptotic stability of fractional discretetime linear systems with delays. Effectiveness of the tests is shown on numerical examples. Keywords: Fractional, Positive, Linear, System, Asymptotic Stability, Test 1. Introduction A dynamical system is called positive if its trajectory starting from any nonnegative initial state remains forever in the positive orthant for all nonnegative inputs. An overview of state of the art in positive theory is given in the monographs [1,2]. Variety of models having positive behavior can be found in engineering, economics, social sciences, biology and medicine, etc. New stability conditions for discretetime linear sys tems have been proposed by M. Busłowicz in [3] and next have been extended to robust stability of fractional dis cretetime linear systems in [4]. The stability of positive continuoustime linear systems with delays has been ad dressed in [5]. The independence of the asymptotic stabil ity of positive 2D linear systems with delays of the num ber and values of the delays has been shown in [6]. The asymptotic stability of positive 2D linear systems without and with delays has been considered in [7,8]. The stability and stabilization of positive fractional linear systems by statefeedbacks have been analyzed in [9,10]. The Hur witz stability of Metzler matrices has been investigated in [11] and some new tests for checking the asymptotic sta bility of positive 1D and 2D linear systems have been proposed in [12]. In this paper new tests for checking asymptotic stability of positive 1D continuoustime and discretetime linear systems without and with delays and of positive 2D linear systems described by the general and the Roesser models will be proposed. It will be shown that the checking of the asymptotic stability of positive 2D linear systems can be reduced to checking of suitable corresponding 1D posi tive linear systems. The paper is organized as follows. In Section 2 new stability tests for positive continuoustime linear systems are proposed. An extension of these tests for positive dis cretetime linear systems is given in Section 3. Applica tion of these tests to checking the asymptotic stability of positive 1D linear systems with delays is given in Section 4. In Section 5 the tests are applied to positive 2D linear systems described by the general and Roesser models and in Section 6 the fractional discretetime linear systems with delays. Concluding remarks are given in Section 7. The following notation will be used: —the set of real numbers, nm —the set of real matrices, mn nm —the set of nm matrices with nonnegative en tries and 1nn , n —the set of Metzler matrices (real matrices with nonnegative offdiagonal entries), —the nn n Inn identity matrix. 2. ContinuousTime Linear Systems Consider the continuoustime linear system () () tAxt (2.1) where () n xt is the state vector and nn A . *This work was supported by Ministry of Science and Higher Educa tion in Poland under work No. N N514 6389 40. The system (2.1) is called (internally) positive if
T. KACZOREK 262 () n xt (0)xx , for any initial conditions [1, 2]. 0t n 0 Theorem 2.1. [1,2] The system (2.1) is positive if and only if A is a Metzler matrix. The positive system is called asymptotically stable if 0 lim( )lim0 At tt xte x for all 0 n x Theorem 2.2. [1,2] The positive system (2.1) is as ymptotically stable if and only if all principal minors of the matrix –A are positive, i.e. ,1,, iin 111 11 12 2 21 22 0, 0, det[] 0 n a aa aa A (2.2) Theorem 2.3. [1,2] The positive system (2.1) is as ymptotically stable only if all diagonal entries of the ma trix A are negative. Let be a Metzler matrix with nega tive diagonal entries (). Let define [] nn ij Aa ii 0,1,,ai n (0) (0) 111, (0) (0) (0) 11 1 (0) (0) (0) (0) 11 ,1 , (0) (0) 22 2, (0) 1 (0) (0) ,2 , (0) 21 (0)(0)(0) (0) 1121, 1 (0) ,1 ... ... , ... ... ... , ... [ ... ], n n n nn nnn n n nnn nnn n aa ab AA cA aa aa A aa a ba ac a (2.3a) and () () 1, 11, (1)(1) ()( 1) (1) () () 1, 1 ,1 , () () 1, 11 () () 11 () () 2, 22, () 1 ... ... ... , ... ... kk kk kn kk kk nk nk nk nkkkk kk nk nn kk kk nk kk nk nk kk kk kn k nk n aa cb AA aaa ab cA aa A a () () ,2 , () 2, 1 ()()() () 11,21,1 () ,1 , ... [...], kk knn k kk kk kk nkkkkn nkk nk a a ba ac a (2.3b) for k = 1,,n – 1. Let us denote by the following elemen tary column operation on the matrix A: addition to the ith column the jth column multiplied by a scalar c. It is wellknown that using these elementary operations we may reduce the matrix [Rij c 11 121, 21 222, ,1 ,2, ... ... ... ... n n nn nn aa a aa a A aa a (2.4) to the lower triangular form 11 21 22 ,1 ,2, 0...0 ... 0 ... nn nn a aa A aaa . (2.5) The reduction of the matrix (2.4) to the form (2.5) is equivalent to postmultiplication of the matrix (2.4) by the upper triangular matrix of the elementary column opera tions of the form 12 1, 2, 1 ... 0 1... 0 0...1 n n rr r R (2.6) i.e. AR (2.7) Note that to reduce to zero the entries 12 of the matrix (2.4) we postmultiply it by the matrix 1 ,,n aa 1, 12 11 11 1 1... 01...0 00... 1 n a a aa R (2.8) and we obtain 11 21 222, 1 ,1 ,2, 0...0 ... ... ... n nn nn a aa a AAR aa a (2.9) where 21 1, 12 21 22 222,2, 11 11 ,1 12,1 1, ,2 ,2,, 11 11 ,, , ,, . n nn nn nn nnnn aa aa aaa a aa aa aa aa aa aa n Next we postmultiply the matrix (2.9) by the matrix 2, 23 22 22 2 10 0...0 01 ... 00 1...0 00 0...1 n a a aa R . (2.10) ] C opyright © 2011 SciRes. CS
T. KACZOREK263 In a similar way we define matrices 3. The matrix (2.6) is the product of the elementary column op erations matrices , i.e. . ,,n RR ,,,RRR 1212 n It is easy to show that if the matrix (2.4) is Metzler ma trix with negative diagonal entries then the matrix (2.5) is also a Metzler matrix. ,,,n RR RR Theorem 2.4. The positive systems with the matrix (2.5) is asymptotically stable if and only if all diagonal entries of the matrix are negative. Proof. The eigenvalues of the matrix (2.5) are equal to its diagonal entries 11 and the positive system is asymptotically stable if and only if all the diagonal entries are negative. ,, nn aa Theorem 2.5. The positive continuoustime linear sys tem (2.1) is asymptotically stable if and only if one of the equivalent conditions is satisfied: 1) the diagonal entries of the matrices defined by (2.3) ()k nk for k = 1,,n – 1 (2.11) are negative, 2) the diagonal entries of the lower triangular matrix (2.5) are negative, i.e. 0 kk a for k = 1,,n (2.12) Proof. Let 1q 1,, ,, q ii j A i be the (q) minor of the matrix A obtained by the deleting all rows except the rows and all columns except the columns 1. In a similar way we define the minors of the matrices qqn 1,,q i q j,,j and R. Applying the CauchyBinet theorem to (2.7) we obtain 11 11 1 ,,,, ,, ,,,, ,, 1 qq q q iiii kk ii kkii kkn AA 1 1 q qq R (2.13) From the structure of the matrix (2.6) it follows that 1 1 ,, 11 ,, 11 ,, 1for ,, 0for q q kk qq ii qq ki ki Rki ki (2.14) Taking into account (2.14) from (2.13) we obtain 11 11 ,, ,, ,, ,, qq qq ii ii ii ii AA n for (2.15) 1, ,q From (2.15) follows the equivalence of the conditions (2.2) and (2.11). To show the equivalence of the condi tions (2.11) and (2.12) note that the computation of the matrix (1) 1n by the use of (2.3b) for k = 1 is equivalent to the reduction to zero of the entries ,,2,, ij aj n of the matrix (2.4) by elementary column operations since 22 2,21 (1) 1 11 ,2 ,,1 ... 1 ... ... ... n n nnn n aa a A a aa a 1 21,n a a . (2.16) Note that 1, 11 0 i a a for i = 2,,n and 1, ,1 11 0 ii aa a for i = 2,,n since and for 11 0a,ij a0i Thus, the matrix (1) 1n is a Metzler matrix. Continuing this procedure after n steps we obtain the Metzler lower triangular matrix (2.5). Therefore, the conditions (2.11) and (2.12) are equivalent. Example 2.1. Consider the positive system (2.1) with the matrix 21 0 011 11 2 A . (2.17) Check the asymptotic stability using the conditions (2.11) and (2.12). Using (2.3) for (2.17) we obtain (0) (0) (1)(0) 22 22 (0) 33 (1) (1) (2)(1) 11 11 (1) 22 11 011 1[1 0] 12 11.52 2 1.5 20.5 1 . bc AA a bc AA a (2.18) The conditions (2.11) of Theorem 2.5 are satisfied and the positive system is asymptotically stable. Using the elementary column operations to the matrix (2.17) we obtain [21 0.5] [32 1] 21 0200 011 011 11 211.52 20 0 010 11.5 0.5 R R A (2.19) The conditions (2.12) of Theorem 2.5 are also satisfied and the positive system is asymptotically stable. 3. DiscreteTime Linear Systems Consider the discretetime linear system 1,{0,1 ii xAxiZ ,} (3.1) where n i x is the state vector and nn A . The system (3.1) is called (internally) positive if n i x , iZ for any initial conditions 0 n x . Theorem 3.1. [1,2] The system (3.1) is positive if and only if nn A . The positive system is called asymptotically stable if 0 lim lim0 i i ii xAx for all 0 n x . From Theorem 2.2 and 3.1 it follows that the nonnega tive matrix is asymptotically stable if and only if the Metzler matrix n I is asymptotically stable. j . Theorem 3.2. [1,2] The positive system (3.1) is as Copyright © 2011 SciRes. CS
T. KACZOREK 264 n ymptotically stable if and only if all principal minors of the matrix ˆ,1,, iiˆˆ [] nn nij AIA a are positive, i.e. 11 12 111 2 21 22 ˆˆ ˆ ˆˆ ˆ ˆ0,0,,det[ ]0 ˆˆ n aa a aa A . Theorem 3.3. [1,2] The positive system (3.1) is as ymptotically stable only if all diagonal entries of the ma trix are less than 1. It is assumed that 1,1, , ii ai n of the matrix [] nn ij Aa since otherwise by Theorem 3.3 the system is unstable. Using (2.3) in a similar way as for the matrix A we define for the matrix ˆˆ [] nij AI a the matrices () ˆk nk for . Using the elemen tary column operations we reduce the matrix 0,1,,1knˆ to the lower triangular form 11 21 22 ,1 ,2, '0 ... 0 ''...0 ' ''... ' nn nn a aa A aa a . (3.4) Theorem 3.4. The positive discretetime system with the matrix (3.4) is asymptotically stable if and only if all diagonal entries of the matrix ˆ' are less than 1. Proof is similar to the proof of Theorem 2.4. Theorem 3.5. The positive discretetime linear system (3.1) is asymptotically stable if and only if one of the equivalent conditions is satisfied: 1) the diagonal entries of the matrices () ˆk nk for k = 1,, n – 1 (3.5) are negative, 2) the diagonal entries of the lower triangular matrix (3.4) are negative, i.e. ' kk a 0 for k = 1,,n. (3.6) Proof. The positive discretetime system (3.1) is as ymptotically stable if and only if the corresponding con tinuoustime system with the Metzler matrix ˆn AI is asymptotically stable. By Theorem 2.5 the positive discretetime system (3.1) is asymptotically stable if one of its conditions is satisfied. □ Example 3.1. Check the asymptotic stability of the positive system (3.1) with the matrix 0.5 0.1 0.2 0.4 A . (3.7) In this case 0.5 0.1 ˆ 0.2 0.6 n AAI . (3.8) Using (3.5) for n = 2 we obtain (1) 12 21 122 11 ˆˆ 0.1 0.2 ˆˆ0.60.560 ˆ0.5 aa Aa a . Condition i) of Theorem 3.5 is satisfied and the posi tive system (3.1) with (3.7) is asymptotically stable. Similarly, using the elementary column operations to the matrix (3.8) we obtain 210.2 0.5 0.10.50 ˆ 0.2 0.60.20.56 R A k . The condition ii) of Theorem 3.5 is also satisfied and the positive system is asymptotically stable. 4. Linear Systems with Delays Consider the continuoustime linear system with q delays [5] 0 1 ()()() q k k tAxt Axtd (4.1) where () n xt ,0 nn is the state vector, k,1,, kq and are delays. 0,1,, k dkq The initial conditions for (4.1) have the form 0 () () txt for [,0td] , . (4.2) max k k dd The system (4.1) is called (internally) positive if () n xt , for any initial conditions 0t0() n xt . Theorem 4.1. The system (4.1) is positive if and only if 0n M and , (4.3) nn k A 1, ,kq where Mn is the set of nn Metzler matrices. Proof is given in [5]. Theorem 4.2. The positive system with delays (4.1) is asymptotically stable if and only if the positive system without delays Ax , 0 q k kn AM (4.4) is asymptotically stable. Proof is given in [5]. To check the asymptotic stability of the system (4.1) Theorem 2.5 is recommended. The application of Theo rem 2.5 to checking the asymptotic stability of the system (4.1) will be demonstrated on the following example. Example 4.1. Consider the system (4.1) with q = 1 and the matrices 01 10.2 0.50.1 , 0.21.40.2 0.8 AA . (4.5) The matrix of the positive system (4.4) without delays has the form C opyright © 2011 SciRes. CS
T. KACZOREK265 201 0.5 0.3 0.4 0.6 AA M . (4.6) Using (2.3) for the matrix (4.6) we obtain (1) 1 0.4 0.3 ˆ0.60.36 0 0.5 A . (4.7) Condition i) of Theorem 2.5 is satisfied and the posi tive system (4.1) with (4.5) is asymptotically stable. Using the elementary column operations to the matrix (4.6) we obtain 3 215 0.5 0.30.50 0.4 0.60.40.36 R A The condition ii) of Theorem 2.5 is also satisfied and the positive system is asymptotically stable. Now let us consider the discretetime linear system with q delays [3] 1 0 , q ikik k Axi Z (4.8) where is the state vector and n i x nn k A , k = 0,1,,q. The initial conditions for (4.8) have the form n k x for k = 0,1,, q. (4.9) The system (4.8) is called (internally) positive if , for any initial conditions n i x iZ n k x for k = 0,1,,q. Theorem 4.3. [2] The system (4.8) is positive if and only if , k = 0,1, ,q. nn k A Theorem 4.4. The positive discretetime system with delays (4.8) is asymptotically stable if and only if the positive system without delays 1ii Ax , 0 q k k A , (4.10) iZ is asymptotically stable. Proof is given in [3]. To check the asymptotic stability of the system (4.8) Theorem 3.5 is recommended. The application of Theo rem 3.5 to checking the asymptotic stability of the system (4.8) will be demonstrated on the following example. Example 4.2. Consider the positive system (4.8) with q = 1 and the matrices 01 0.2 0.20.20.1 , 0.1 0.20.1 0.3 AA . (4.11) The matrix of the positive system (4.10) without de lays has the form 01 0.4 0.3 0.2 0.5 AA A In this case 0.6 0.3 ˆ 0.2 0.5 n AAI (4.13) and using the elementary column operation to (4.13) we obtain 210.5 0.6 0.30.60 0.2 0.50.2 0.4 R . The condition ii) of Theorem 3.5 is satisfied and the positive system is asymptotically stable. 5. 2D Linear Systems Consider the general autonomous model of 2D linear systems 1, 10 ,11,2 , 1,, i jiji jij AxAxAxijZ (5.1) where , n ij x is the state vector and nn k A , k = 0,1,2. Boundary conditions for (5.1) have the form ,0 n i x , iZ and , 0, n j x jZ . (5.2) The model (5.1) is called (internally) positive if , n ij x , ,ij Z for any initial conditions ,0 n i x , iZ , 0, n j x , jZ . Theorem 5.1. [2] The system (5.1) is positive if and only if nn k A , k = 0,1,2. (5.3) The Roesser autonomous model of 2D linear systems has the form [2] 11 12 1, , 21 22 ,1 , ,, hh ij ij vv ij ij AA xx ij Z AA xx (5.4) where 1 , n h ij x and 2 , n v ij x are the horizontal and vertical state vectors at the point (i, j) and ,kl nn kl A , k, l = 1,2. Boundary conditions for (5.4) have the form 1 0, n hj x , jZ and , 2 ,0 n v i x iZ . (5.5) The model (5.4) is called (internally) positive if 1 , n h ij x and 2 , n v ij x for any initial conditions 1 0, n hj x , jZ and , . 2 ,0 n v ix Theorem 5.2. [2] The Roesser model (5.4) is positive if and only if iZ 11 12 21 22 nn AA AA , . (5.6) 1 nnn 2 The positive general model (5.1) is called asymptoti cally stable if . (4.12) , , lim 0 ij ij for all , ,0 n i x iZ Copyright © 2011 SciRes. CS
T. KACZOREK 266 and , . (5.7) 0, n j x jZ Similarly, the positive Roesser model (5.4) is called asymptotically stable if , ,, lim 0 h ij v ij ij x x for all , 1 0, n hj x jZ and 2 ,0 n v i x , . (5.8) iZ Theorem 5.3. The positive general model (5.1) is as ymptotically stable if and only if the positive 1D system 1012 , ii , AxAAAAi Z (5.9) is asymptotically stable. Proof is given in [8, 6]. Theorem 5.4. The positive Roesser model (5.4) is as ymptotically stable if and only if the positive 1D system 11 12 1 21 22 , ii AA xiZ AA (5.10) is asymptotically stable. Proof is given in [8, 6]. To check the asymptotic stability of the positive gen eral model (5.1) and the positive Roesser model (5.4) the Theorem 3.5 is recommended. The application of Theo rem 3.5 to checking the asymptotic stability of the models (5.1) and (5.4) will be demonstrated on the following examples. Example 5.1. Consider the positive general model (5.1) with the matrix 011 0.10.200.10.20.3 ,, 0.10.1 00.1 0.10.2 AAA . (5.11) In this case 012 0.3 0.6 0.2 0.4 AA AA (5.12) and 0.70.6 ˆ 0.2 0.6 n AAI . (5.13) Using the elementary column operation to (5.13) we obtain 6 217 0.7 0 0.7 0.6 ˆ3 0.2 0.60.27 R A . The condition ii) of Theorem 3.5 is satisfied and the positive general model with (5.11) is asymptotically sta ble. Example 5.2. Consider the positive Roesser model (5.4) with the matrices 11 12 21 22 A A A (5.14a) and 11 12 21 22 0.6 0.20.1 ,, 0.1 0.40.2 [0.2 0.1],[0.8]. AA AA (5.14b) In this case 0.4 0.20.1 ˆ0.10.60.2 0.2 0.10.2 n AAI . (5.15) Using the elementary column operation to (5.15) we obtain 210.5 310.25 22.5 32 55 0.4 0.20.10.400 0.10.60.20.10.55 0.225 0.2 0.10.20.20.20.15 0.4 00 0.1 0.550 0.2 0.20.0682 R R R The condition ii) of Theorem 3.5 is satisfied and the positive Roesser model with (5.14) is asymptotically sta ble. In a similar way as for 1D linear systems using the approach given in [7] the considerations can be easily extended to 2D linear systems with delays and to frac tional 1D and 2D linear systems. 6. Fractional Positive DiscreteTime Linear Systems Consider the autonomous fractional discretetime linear systems with q delays [10] 11 11 10 (1) q ij iij jk kik xA j x 01, (6.1) where is the fractional order, is the state vector, n i x ,0,1,, nn k kq and 1for 0 for1, 2, ( 1)...(1) ! j jj j j (6.2) The fractional system (6.1) is called (internally) posi tive if n i x , for any initial conditions n k x , 0,1, , kq. Theorem 6.1. [10] The fractional system (6.1) is posi tive if and only if for 1 nn kkn AcI 0,1, , kq C opyright © 2011 SciRes. CS
T. KACZOREK267 where (1)k k ck . The fractional positive system (6.1) is called asymp totically stable if lim 0 i ix for all , . (6.3) n k x 0,1,,kq Theorem 6.2. [10] The fractional positive system (6.1) is asymptotically stable if and only if the positive dis cretetime system without delays 1ii Ax , 0 q n kk I A (6.4) is asymptotically stable. Proof is given in [10]. To check the asymptotic stability of the fractional posi tive system (6.1) Theorem 3.5 is recommended. The ap plication of Theorem 3.5 to checking the asymptotic sta bility of the system (6.1) will be demonstrated on the fol lowing example. Example 6.1. Consider the fractional system (6.1) for 5.0 with q = 1 and the matrices 01 0.550.10.20.1 , 0.050.50.050.2 AA . (6.5) The fractional system is positive since 22 012 02 0.05 0.1 0.05 0 AcIAI (6.6a) and 22 122 22 0.075 0.1 (1) 0.05 0.075 2 AcI AI (6.6b) Therefore, to check the asymptotic stability of the positive system we may use Theorem 3.5. Using (3.5) for n = 2 and (6.4) we obtain 2012 1.25 0.2 ˆ20.1 1.3 AAIA AI (6.7) and (1) 12 21 122 11 ˆˆ 0.2 0.1 ˆˆ1.31.284 0 ˆ1.25 aa Aa a . (6.8) The condition i) of Theorem 3.5 is also satisfied and the positive system is asymptotically stable. Using the elementary column operations to the matrix (6.7) we obtain 210.16 1.25 0.21.250 0.1 1.30.1 1.284 R . The condition ii) of Theorem 3.5 is satisfied and the positive system is asymptotically stable. This approach can be also applied to checking the asymptotic stability of the positive 2D linear systems with delays. 7. Concluding Remarks New tests for checking asymptotic stability of positive 1D continuoustime and discretetime linear systems without and with delays and of positive 2D linear sys tems described by the general and the Roesser models have been proposed. The tests are based on the Theorem 2.5 and Theorem 3.5. Checking of the asymptotic stabil ity of positive 2D linear systems has been reduced to checking of suitable corresponding 1D positive linear systems. It has been shown that the stability tests can be also applied to checking the asymptotic stability of fractional discretetime linear systems with delay. The tests can be also extended to 2D continuousdiscrete lin ear systems and to 1D and 2D fractional linear systems. An open problem is extension of these considerations to 2D positive switched linear systems. 8. References [1] L. Farina and S. Rinaldi, “Positive Linear Systems; Theory and Applications,” John Wiley & Sons, Hoboken, 2000. doi:10.1002/9781118033029 [2] T. Kaczorek, “Positive 1D and 2D Systems,” Springer Verlag, London, 2002. doi:10.1007/9781447102212 [3] M. Busłowicz, “Simple Stability Conditions for Linear Systems with Delays,” Bulletin of the Polish Academy of Sciences, Vol. 56, No. 4, 2008, 319324. [4] M. Busłowicz, “Robust Stability of Positive Discrete Time Linear Systems of Fractional Order,” Bulletin of the Polish Academy of Sciences, Vol. 58, No. 4, 2010, 567572. doi:10.2478/v1017501000578 [5] T. Kaczorek, “Stability of Positive ContinuousTime Linear Systems with Delays,” Bulletin of the Polish Academy of Sciences, Vol. 57, No. 4, 2009, 395398. doi:10.2478/v101750100143y [6] T. Kaczorek, “Independence of Asymptotic Stability of Positive 2D Linear Systems with Delays of Their Delays,” International Journal of Applied Mathematics and Computer Science, Vol. 19, No. 2, 2009, 255261. doi:10.2478/v1000600900217 [7] T. Kaczorek, “Asymptotic Stability of Positive 2D Linear Systems with Delays,” Bulletin of the Polish Academy of Sciences, Vol. 57, No. 2, 2009, 133138. doi:10.2478/v1017501001134 [8] T. Kaczorek, “Asymptotic Stability of Positive 2D Linear Systems,” Computer Applications in Electrical Engi neering, Poznan University of Technology, Electrical En gineering Committee of Polish Academy of Sciences, IEEE Poland Section, Poznan. Copyright © 2011 SciRes. CS
T. KACZOREK Copyright © 2011 SciRes. CS 268 [9] T. Kaczorek, “Stability and Stabilization of Positive Frac tional Linear Systems by StateFeedbacks,” Bulletin of the Polish Academy of Sciences, Vol. 58, No. 4, 2010, 517554. [10] T. Kaczorek, “Selected Problems of Fractional System Theory,” Springer Verlag, London, 2011. doi:10.1007/9783642205026 [11] K. S. Narendra and R. Shorten, “Hurwitz Stability of Me tzler Matrices,” IEEE Transactions on Automatic Control, Vol. 55, no. 6 June 2010, pp. 14841487. doi:10.1109/TAC.2010.2045694 [12] T. Kaczorek, “New Stability Tests of Positive 1D and 2D Linear Systems,” Proceeding of 25th European Confer ence Modelling and Simulation, Krakow, 710 June 2011.
