Stability Analysis of Positive Linear Systems by Decomposition of the State Matrices Into Symmetrical and Antisymmetrical Parts

BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 67, No. 4, 2019 DOI: 10.24425/bpasts.2019.130185

Stability analysis of positive linear systems by decomposition of the state matrices into symmetrical and antisymmetrical parts

T. KACZOREK* Bialystok University of Technology, Faculty of Electrical Engineering, Wiejska 45D, 15-351 Bialystok

Abstract. The stability of positive linear continuous-time and discrete-time systems is analyzed by the use of the decomposition of the state matrices into symmetrical and antisymmetrical parts. It is shown that: 1) The state Metzler matrix of positive continuous-time linear system is Hurwitz if and only if its symmetrical part is Hurwitz; 2) The state matrix of positive linear discrete-time system is Schur if and only if its symmetrical part is Hurwitz. These results are extended to inverse matrices of the state matrices of the positive linear systems.

Key words: linear, positive, system, decomposition, state matrix, stability.

1. Introduction The following notation will be used: ℜ – the set of real n m n m numbers, ℜ £ – the set of n m real matrices, ℜ+£ – the set £ n n 1 A dynamical system is called positive if its trajectory starting of n m real matrices with nonnegative entries and ℜ = ℜ £ , £ + + from any nonnegative initial state remains forever in the posi- M – the set of n n Metzler matrices (real matrices with non- n £ tive orthant for all nonnegative inputs. An overview of state of negative off-diagonal entries), I – the n n identity matrix. n £ the art in positive systems theory is given in the monographs [1, 3, 7]. A variety of models having positive behavior can be found in engineering, especially in electrical circuits [15], eco- 2. Preliminaries nomics, social sciences, biology and medicine, etc. [3, 7]. The positivity and stability of linear systems have been Consider the continuous-time linear system investigated in [2, 5‒6, 8, 9, 16, 17, 22, 23] and of nonlinear systems in [10, 11]. A comparison of the stability of dis- x ̇ = Ax, (1) crete-time and continuous-time linear systems has been given n n n in [4]. The stability of interval positive linear systems with state where x = x(t) ℜ is the state vector and A = a ℜ £ . 2 ij 2 matrices in integer and rational powers has been analyzed in £ ¤ [14]. The linear systems have been intensively investigated in Definition 1. [3, 7] The system (1) is called (internally) positive [11, 12‒15, 18‒20] and the descriptor fractional linear systems if x(t) ℜn, t 0 for all x(0) ℜn. 2 + ¸ 2 + with different fractional orders in [21]. In this paper the asymptotic stability of positive continu- Theorem 1. [3, 7] The system (1) is positive if and only if ous-time and discrete-time linear systems by the decomposition of the state matrices into symmetrical and antisymmetrical parts A Mn. (2) will be addressed. 2 The paper is organized as follows. In Section 2 some prelim- Definition 2. [3, 7] The positive system (1) is called asymptot- inaries concerning positive continuous-time and discrete-time ically stable (the matrix A is called Hurwitz) if linear systems are recalled. Decomposition of the state matrices of positive linear continuous-time and discrete-time systems in n lim x(t) = 0 for all x(0) ℜ . (3) → + symmetrical and antisymmetrical parts and the stability of the t 1 2 symmetrical parts are addressed in Section 3. The Comparison of the stability of positive continuous-time and discrete-time Theorem 2. [3, 7] The positive system (1) is asymptotically linear systems is presented in Section 4. Some concluding stable if and only if one of the following equivalent conditions remarks are given in Section 5. is satisfied: 1. All coefficients of the characteristic polynomial

* n n 1 e-mail: [email protected] det In s A = s + an 1 s ¡ + … + a1s + a0 (4) ¡ ¡ Manuscript submitted 2019-03-19, revised 2019-05-01 and 2019-05-23, £ ¤ initially accepted for publication 2019-06-19, published in August 2019 are positive, i.e. a > 0 for i = 0, 1, …, n 1. i ¡

Bull. Pol. Ac.: Tech. 67(4) 2019 761 T. Kaczorek

– n n 2. All principal minors M , i = 1, …, n of the matrix –A are Theorem 6. [15] If A ℜ £ is Schur matrix then i 2 + positive, i.e. –1 n n (I A) ℜ £ . (14) – – – a11 – a12 n ¡ 2 + M1 = – a11 > 0, M2 = > 0, …, j j – a21 – a22 (5) Theorem 7. Let sk, k = 1, …, n be the eigenvalues of the matrix – n n M = det – A > 0. A ℜ £ . Then: n j j 2 1) –sk, k = 1, …, n are the eigenvalues of the matrix –A, £ ¤ T –1 3. There exists strictly positive vector λ = λ1 λn , λk > 0, 2) sk , k = 1, …, n are the eigenvalues of the inverse matrix ¢¢¢ –1 n n k = 1, …, n such that (det A = 0) A ℜ £ . £ ¤ 6 2 Aλ < 0 and λTA < 0. (6) Proof. From the equality

Theorem 3. [15] If A M is asymptotically stable then –I I λ A = I (–λ) (– A) (15) 2 n n n ¡ n ¡ –1 n n £ ¤ £ ¤ –A ℜ £ . (7) it follows that det I (–λ) (– A) = 0 if and only if det I λ + n ¡ n ¡ 2 A = 0 since det – I (I λ A) = (–1)n det I λ A . ¡ £ n n ¡ ¤ n ¡ £ Consider the discrete-time linear system If det A = 0 then from the equality ¤ 6 £ ¤ £ ¤ x = Ax , i = 0, 1, … (8) –(Aλ)–1 I λ A = I λ–1 A–1 (16) i+1 1 n ¡ n ¡ n n –1 –1 where x is the state vector and A = a ℜ £ . it follows that det £I λ A¤ =£ 0 if and only¤ if det I λ i ij 2 n ¡ n ¡ A = 0 since £ ¤ ¡ £ ¤ £ Definition 3. [3, 7] The system (8) is called (internally) positive (Aλ)–1( λ A) = (Aλ)–1 λ A n n ¤det – In det – det In if xi ℜ+, i 0 for all x0 ℜ+. ¡ ¡ 2 ¸ 2 –1 and det Aλ = 0. □ £ 6 ¤ £ ¤ £ ¤ Theorem 4. The system (8) is positive if and only if £ ¤ Example 1. The matrix n n A ℜ £ . (9) 2 + –2 1 Definition 4. [3, 7] The positive system (8) is called asymptot- A = (17) ically stable (the matrix A is called Schur) if 1 –2

lim x = 0 for all x ℜn . (10) with characteristic polynomial → i 0 + t 1 2

λ + 2 –1 2 Theorem 5. [3, 7] The positive system (8) is asymptotically det I2λ A = = λ + 4λ + 3 (18) stable if and only if one of the following equivalent conditions ¡ –1 λ + 2 £ ¤ is satisfied: j j 1) All coefficients of the characteristic polynomial has the eigenvalues λ1 = –1, λ2 = –3. The matrix n n 1 det In(z + 1) A = z + an 1 z ¡ + … + ¡ ¡ (11) 2 –1 det£In(z + 1) A¤ + a1z + a0 – A = (19) ¡ –1 2 £ ¤ are positive, i.e. a > 0 for i = 0, 1, …, n 1. i ¡ 2) All principal minors M ̂ , i = 1, …, n of the matrix I A with characteristic polynomial i n ¡ are positive, i.e. λ 2 –1 1 a – a 2 ̂ ̂ 11 12 det I2λ (– A) = ¡ = λ + 4λ + 3 (20) M1 = 1 a11 > 0, M2 = ¡ > 0, ¡ –1 λ 2 j ¡ j – a21 1 a22 ¡ (12) £ ¤ ¡ ̂ j j …, Mn = det In A > 0. j j ¡ has the eigenvalues λ1 = 1, λ2 = 3. £ ¤ The inverse matrix of (17) has the form 3) There exists strictly positive vector λ = λ λ T, λ > 0, 1 ¢¢¢ n k k = 1, …, n such that £ ¤ 1 –2 –1 A–1 = (21) Aλ < λ and λTA < λT. (13) 3 –1 –2

762 Bull. Pol. Ac.: Tech. 67(4) 2019 Stability analysis of positive linear systems by decomposition of the state matrices into symmetrical and antisymmetrical parts

and its characteristic polynomial –9 + 33 –9 33 Note that the eigenvalues s = , s = ¡ of 1 2 2 2 – – 2 1 the matrix A and of the matrix As, s1, s2 are different and satisfy λ + the condition trA = trA = –9 = s– + s– . 3 3 4 1 s 1 2 det I λ A–1 = = λ2 + λ + (22) 2 ¡ 1 2 3 3 λ + Theorem 8. The Metzler matrix A Mn is Hurwitz if and only 3 3 2 £ ¤ if its symmetrical part As is Hurwitz.

1 has the zeros λ1 = –1, λ2 = – /3. Proof. By Theorem 2 the matrix A Mn is Hurwitz if and only j j 2 n This confirms the Theorem 7. if there exists a strictly positive vector λ ℜ such that 2 + Aλ < 0 and λTA < 0. (26) 3. Decomposition of the state matrices into T symmetrical and antisymmetrical parts Taking into account that (λTA) = ATλ < 0 and (1, 2) we obtain

Consider the Metzler matrix A = a M which is in general ij 2 n A + AT Aλ + ATλ case not symmetrical. It is well-known the this matrix can be A λ = λ = < 0. (27) £ ¤ s 2 2 decomposed into symmetrical part As and the antisymmetrical part Aa Therefore, the Metzler matrix A is Hurwitz if and only if the A = As + Aa (23a) matrix As is Hurwitz. □ where Remark 1. If the matrix A = a M satisfies the conditions ij 2 n T T A + A A A n £ ¤ As = , Aa = ¡ (23b) 2 2 and ∑ aij < 0 for i = 1, …, n j =1 and T denotes the transpose. n (28) The following properties of the matrices (23b) are well- and aij < 0 for j = 1, …, n known: ∑ – – i =1 1) All eigenvalues s1, …, sn of the matrix As are real and of the matrix Aa = a ̂ ij , s ̂ 1, …, s ̂ n are imaginary. then we may choose the strictly positive vector in the form 2) The trace of the matrix A is λ = 1, …, 1 T ℜn . £ ¤ 2 + n n – £ ¤ trA = trAs = ∑ aii = ∑ si (24a) Example 3. The Metzler matrix i =1 i =1 and – 4 2 1 n n A = 3 – 3 0 (29) trAa = ∑ a ̂ ii = ∑ s ̂ i = 0. (24b) i =1 i =1 3 1 – 5

Example 2. The Metzler matrix is Hurwitz since its characteristic polynomial

– 4 4 s + 4 –2 –1 A = M2 (25a) 2 – 5 2 det I3s A = –1 s + 1 0 = ¡ (30) can be decomposed into £ ¤ –3 –1 s + 5 T – 4 3 3 2 A + A det I3s A = js + 12s + 42s + 40j As = = , ¡ 2 3 – 5 £ ¤ (25b) has positive coefficients (Theorem 2). T A AT 0 1 Choosing for the matrix (29) λ = 0.74 0.73 0.70 we Aa = ¡ = . 2 – 1 0 obtain £ ¤

–9 + 37 –9 37 – 4 2 1 0.74 – 0.80 The eigenvalues of A are real s– = , s– = ¡ s 1 2 2 2 ̂ ̂ – – Aλ = 1 – 3 0 0.73 = – 1.45 (31a) and of Aa are imaginary s1 = j, s2 = – j and trAs = s1 + s2 = –9, ̂ ̂ trAa = s1 + s2 = 0. 3 1 – 5 0.70 – 0.55

Bull. Pol. Ac.: Tech. 67(4) 2019 763 T. Kaczorek and Proof. Proof follows immediately from Theorems 8 and the – 4 2 1 decomposition of the matrix into the symmetrical and antisymmetrical parts. □ λTA = 0.74 0.73 0.70 1 – 3 0 = (31b) £ ¤ 3 1 – 5 Example 4. The matrix

T λ A = – 0.13 – 0.01 – 2.76 . 0.4 0.2 A = (36) The symmetrical£ and antisymmetrical¤ parts of the matrix (29) 0.4 0.6 have the forms has the symmetrical part – 4 1.5 2 A + AT T As = = 1.5 – 3 0.5 , A + A 0.4 0.3 1 0 2 Ads = I2 = = 2 0.5 – 5 2 ¡ 0.3 0.6 ¡ 0 1 (32) (37a) 0 0.5 –1 – 0.6 0.3 A = A AT ds A = ¡ = – 0.5 0 – 0.5 . 0.3 – 0.5 a 2 1 0.5 0 and the antisymmetrical part

λ = T The symmetrical part As is also Hurwitz since for 1 1 1 A AT 0 – 0.1 A = ¡ = . (37b) £ ¤ da 2 – 4 1.5 2 1 – 0.5 0.1 0 λ = = As 1.5 – 3 0.5 1 –1 (33a) The matrix (37a) is Hurwitz Metzler matrix. 2 0.5 – 5 1 – 2.5 Example 5. The matrix and its characteristic polynomial 0.4 0.1 0.2 s + 4 –1.5 –2 A = 0 0.2 0.1 (38) det I3s A = –1.5 s + 3 – 0.5 = ¡ (33b) 0.1 0.3 0.5 £ ¤ –2 – 0.5 s + 5

3 2 det I3s A = js + 12s + 40.5s + 32.75j is Schur matrix since the characteristic polynomial ¡ £ ¤ has positive coefficients. z + 0.6 – 0.1 – 0.2 Note that the matrix (29) and its symmetrical part As have different characteristic polynomials. det I3(z + 1) A = 0 z + 0.8 – 0.1 = To extend Theorem 8 to the discrete-time linear system we ¡ (39) decomposed the state matrix of the system (8) into the sym- £ ¤ – 0.1 – 0.3 z + 0.5 metrical part 3 2 det I3(z + 1) A = jz + 1.9z + 1.13z + 0.205j ¡ A + AT A I + AT I A = d d = ¡ n ¡ n = A I , £ ¤ ds 2 2 s ¡ n (34) of the matrix A = I3 has the positive coefficients (condition 1 T A + A of Theorem 5). The£ ¤same result we obtain for the matrix (38) A = T s 2 using the condition 3 of Theorem 5 since for λ = 1 1 1 £ ¤ and the antisymmetrical part 0.4 0.1 0.2 1 0.7 1

A AT A I AT + I A AT 0 0.2 0.1 1 = 0.3 < 1 . (40) A = d ¡ d = ¡ n ¡ n = ¡ . (35) da 2 2 2 0.1 0.3 0.5 1 0.9 1

Theorem 9. The matrix A of the discrete-time system (8) is Decomposition of the matrix A = A I for (34) in the d ¡ n Schur matrix if and only if the matrix A I is Hurwitz. symmetrized and antisymmetrical parts yields s ¡

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0.4 0.05 0.15 is Hurwitz since its characteristic polynomial A + AT Ads = In = 0.05 0.2 0.2 + 2 ¡ ¡ s 4 – 4 2 det I2 s A = = s + 9s + 12 (47) 0.15 0.2 0.5 ¡ – 2 s + 5 (41a) £ ¤ 1 0 0 – 0.6 0.05 0.15 has positive coefficients.j j A 0 1 0 = 0.05 – 0.8 0.2 . The inverse matrix of (46) ds ¡ 0 0 1 0.15 0.2 – 0.5 5 1 –1 – – – 4 4 12 3 A–1 = = (48) 1 1 0 0.05 0.05 2 – 5 – – A AT 6 3 A = ¡ I = – 0.05 0 – 0.1 . (41b) da 2 ¡ n – 0.05 0.1 0 –1 2 2 has all negative coefficients and – A ℜ+£ . T 2 Note that for λ = 1 0.8 we have The matrix (41a) is Hurwitz Metzler matrix and the matrix (38) £ ¤ is Schur matrix. – 4 4 1 – 0.8 Now we shall extend the decomposition into the symmetri- Aλ = = < 0, (49a) cal and the antisymmetrical parts the inverse matrix A–1 to the 2 – 5 0.8 –2 Metzler Hurwitz matrix A M 2 n –1 5 1 41 A = U + V, (42a) – – – 12 3 1 60 A–1λ = = < 0 (49b) where 1 1 23 – – 0.8 – A–1 + (A–1)T A–1 (A–1)T 6 3 30 U = , V = ¡ . (42b) 2 2 and

Theorem 10. Let A Mn be Hurwitz. Then 5 1 2 – – T –1 12 3 23 9 –1 n λ A = 1 0.8 = – – < 0. (49c) As λ < 0 for any strictly positive λ ℜ+ . (43) 1 1 2 – – ∙ 60 15 ¸ £ ¤ 6 3 –1 n n –2 –1 –1 Proof. By Theorem 3 – A ℜ+£ and A = (– A )(– A ) n n 2 ℜ £ . The symmetrical and antisymmetrical parts of (48) have the 2 + Premultiplying Aλ < 0 by the matrix A–2 we obtain forms

–2 –1 5 1 A Aλ = A λ < 0. (44a) – – –1 + –1 T –1 A (A ) 12 2 T –2 n n As = = , Similarly, postmultiplying λ A < 0 by A ℜ+£ we obtain 2 1 1 2 – – 6 3 λTAA–2 < 0 and λTA–1 < 0. (44b) (50) 1 T 0 Therefore, using (44) we obtain A–1 (A–1) 6 A–1 = ¡ = . a 2 1 – 0 A–1 + (A–1)T 6 Uλ = λ < 0. (45) 2 and and the condition (43) is satisfied. □ 5 1 41 – – – 12 2 1 60 Example 6. The Metzler matrix A–1λ = = < 0. (51) s 1 1 23 – – 0.8 – 2 3 30 – 4 4 A = (46) 2 – 5 Therefore, the relationship (51) confirms the Theorem 10.

Bull. Pol. Ac.: Tech. 67(4) 2019 765 T. Kaczorek

(m 1) (m 1) 4. Comparison of the stability of positive By assumption if Am 1 ℜ+ ¡ £ ¡ and satisfies (52) then –1 ¡ 2 –1 continuous-time and discrete-time linear systems – Am 1 Mm 1 is Hurwitz and from (56) we have – Am Mm ¡ 2 ¡ m 1 1 1 m 1 2 is Hurwitz since A12 ℜ+ ¡ £ , A21 ℜ+£ ¡ and am = amm –1 2 2 ¡ Theorem 11. If the eigenvalues λ k, k = 1, …, n are the eigen- A21Am 1 A12 > 0. n n ¡ ¡ values of the matrix A ℜ £ satisfying the condition This completes the proof. □ 2 +

Reλk > 0 for k = 1, …, n. (52) Example 7. The matrix then the matrix 2 1 3 3 3 –1 A = 2 3 5 ℜ+£ (57) –A Mn (53) 2 2 2 0 2 is Hurwitz. has the characteristic polynomial Proof. Proof will be accomplished by induction with respect to n. For n = 1 the hypothesis is evident since A = a > 0 and is s 2 –1 –3 11 ¡ – A–1 = – 1/ M Hurwitz. a1 2 1 det I s A = –2 s 3 – 5 = a a 3 ¡ ¡ 11 12 2 2 (58) For n = 2 we have A2 = ℜ+£ , aij 0, a11 > 0, £ ¤ 0 0 s 2 a21 a22 2 ¸ ¡ a > 0 since λ > 0, k = 1, …, n implies det A = a a 3 2 22 k 11 22 ¡ det I3s A = js 7s + 14s 8 j a12 a21 = λ1λ2 > 0 and ¡ ¡ ¡ ¡ £ ¤ and its zeros s1 = 1, s2 = 2, s3 = 4 satisfy the condition (52). –1 1 a22 – a12 –A2 = = The inverse matrix of (57) has the form det A – a21 a11 (54) 3 1 1 –1 1 – a22 a12 – – – –A2 = M2 4 4 2 det A a – a 2 21 11 1 1 1 A–1 = – – – (59) –1 2 2 2 is Hurwitz (Re(– λk) < 0, k = 1, 2) m m Assume that A ℜ £ , m = 2, 3, …, n satisfies the con- 1 m 2 + – 0 – 0 – dition (52). Using the extension method we shall show that 2 – A–1 M is Hurwitz. It is easy to verify that if m 2 m and its characteristic polynomial a11 a12 … a1m 3 1 1 a21 a22 … a2m Am 1 A12 ¡ s Am = = , ¡ 4 4 2 … A21 amm –1 1 1 1 det I3s A = s = am1 am2 … amm ¡ 2 ¡ 2 2 (60) (55) £ ¤ 1 a 0 0 s 1m ¡ 2 a 2m 7 7 1 A12 = , A21 = am1 am2 … am, m 1 det I s A–1 = s3 s2 + s ¡ 3 ¡ ¡ 8 4 ¡ 8 £ ¤ ¤ am 1, m £ ¡ j j –1 –1 1 –1 1 has the zeros s1 = 1, s2 = /2, s3 = /4. then From (59) we have the matrix

–1 –1 –1 –1 Am 1A12 A21Am 1 Am 1A12 3 1 1 Am 1 + ¡ ¡ – ¡ – – – ¡ am am 4 4 2 A–1 = , m –1 A21Am 1 1 (56) –1 1 1 1 ¡ – A = – – – M (61) – 2 2 2 2 3 am am 1 –1 – 0 – 0 – am = amm A21 Am 1 A12 . 2 ¡ ¡

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̂ = ̂ = 1/ ̂ = 1/ 5 1 with the eigenvalues s1 –1, s2 – 2, s3 – 4. Therefore, s – the matrix (61) is Hurwitz Metzler. ¡ 12 3 3 1 det I s A–1 = = s2 + – s + (66) 2 ¡ 1 1 4 12 Let the continuous-time linear system (1) with – s Theorem 12. £ ¤ 6 ¡ 3 the matrix A M be positive and Hurwitz. Then: 2 n 1) The system with the matrix – A / M is not positive and has one negative coefficient. 2 n not Hurwitz; j j –1 2) The system with the matrix A / Mn is not positive but Theorem 13. Let the discrete-time linear system (8) with the 2 n n Hurwitz; matrix A ℜ+£ be positive and Schur. Then: –1 2 n n 3) The system with the matrix – A M is positive and not 1) The system with the matrix – A / ℜ £ is not positive but 2 n 2 + Hurwitz. the matrix is Schur; –1 n n 2) The system with the matrix A / ℜ £ is not positive and 2 + Proof. the matrix is not Schur; –1 n n 1) If A M then – A / M (is not a Metzler matrix) and by 3) The system with the matrix – A / ℜ £ is not positive and 2 n 2 n 2 + condition 1 of Theorem 7 is not Hurwitz. the matrix is not Schur. 2) If A M then A–1 / M and by condition 2 of Theorem 7 2 n 2 n is not Hurwitz. Proof. –1 n n n n n n 3) If A M then by Theorem 3 the matrix – A ℜ £ and 1) If A ℜ £ then – A / ℜ £ and the system is not positive. 2 n 2 + 2 + 2 + it is a Metzler matrix. By condition 1) and 2) of Theorem 7 By condition 1 of Theorem 7 if zk is eigenvalue of A then the eigenvalues of the matrix are located in the right-hand – zk is the eigenvalue of – A. The discrete-time system is side of the complex plane and the system is unstable. □ asymptotically stable since the stability of the discrete-time linear systems depends only on the moduli of the eigen- Example 8. (continuation of Example 6). values. n n –1 n n The system with the matrix (46) is positive and Hurwitz and 2) If A ℜ £ then A / ℜ £ and the system is not positive. 2 + 2 + the matrix – A has the form By condition 2 of Theorem 7 if zk is the eigenvalue of A then –1 –1 zk is the eigenvalue of A . The discrete-time system with – 4 – 4 –1 –1 the matrix A is unstable since if zk < 1 then zk > 1. – A = / Mn . (62) –1 – 2 – 5 2 3) The system with the matrix – A j isj not positivej j since –1 n n – A / ℜ £ . The system is unstable since the eigenvalues 2 + The corresponding system to (62) is not positive and unstable of the matrix have moduli greater 1. □ since its characteristic polynomial Example 9. The discrete-time linear system (8) with the matrix

s 4 4 2 det I2 s A = ¡ = s 9s + 12 (63) 0.4 0.1 2 2 ¡ ¡ A = ℜ £ (67) 2 s 5 2 + £ ¤ j ¡ j 0.2 0.3 has one negative coefficient. is positive and the matrix is Schur since the characteristic poly- The inverse matrix (46) has the form (48) and its coeffi- nomial cients are negative. Therefore, it is not a Metzler matrix. The characteristic polynomial of the matrix (48) z + 0.6 – 0.1 det I2(z + 1) A = = ¡ – 0.2 z + 0.7 (68) 5 1 £ ¤ s + det I (z + 1) A = jz2 + 1.3z + 0.4 j 12 3 3 1 2 det I s A–1 = = s2 + s + (64) ¡ 2 ¡ 1 1 4 12 £ ¤ s + has positive coefficients. £ ¤ 6 3 The inverse matrix of (67) has the form has positive coefficients and the matrix is Hurwitz. –1 j j –1 0.4 0.1 –3 –1 2 2 From (48) we have A = = / ℜ+£ (69) 0.2 0.3 –2 –4 2 5 1

12 3 –A–1 = M (65) and the characteristic polynomial 1 1 2 2

6 3 –1 z + 0.6 1 det I2(z + 1) A = = ¡ 2 z 3 (70) The matrix (65) is not Hurwitz since its characteristic poly- £ ¤ ¡ nomial det I (z + 1) A = z2 j 5z + 4 j 2 ¡ ¡ £ ¤ Bull. Pol. Ac.: Tech. 67(4) 2019 767 T. Kaczorek has one negative coefficient. Therefore, the system with the [2] M. Busłowicz and T. Kaczorek, “Simple conditions for practical matrix (69) is not positive and unstable. stability of positive fractional discrete-time linear systems”, Int. From (69) we have J. Appl. Math. Comput. Sci., 19(2), 263‒169 (2009). [3] L. Farina and S. Rinaldi, Positive Linear Systems; Theory and Applications, J. Wiley, New York 2000. –1 –3 –1 2 2 [4] T. Kaczorek, “Analysis and comparison of the stability of dis- –A = / ℜ+£ (71) –2 –4 2 crete-time and continuous-time linear systems”, Archives of Con- trol Sciences, 26(4), 441‒432 (2016). [5] T. Kaczorek, “Analysis of positivity and stability of discrete-time and the characteristic polynomial and continuous-time nonlinear systems”, Computational Prob- lems of Electrical Engineering, 5(1), 11‒16 (2015). [6] T. Kaczorek, “Fractional positive continuous-time linear systems –1 z + 3 –1 2 det I2z + A = = z + 7z + 10 (72) and their reachability”, Int. J. Appl. Math. Comput. Sci., 18(2), –2 z + 4 223‒228 (2008). £ ¤ j j [7] T. Kaczorek, Positive 1D and 2D Systems, Springer-Verlag, Lon- don 2002. has the zeros z1 = – 2, z1 = – 5. Therefore, the system with the matrix (71) is not positive and unstable. [8] T. Kaczorek, “Positive fractional continuous-time linear systems A The above considerations can be extended to linear systems with singular pencils”, Bull. Pol. c.: Tech., 60(1), 9‒12 (2012). [9] T. Kaczorek, “Positive singular discrete-time linear systems”, with state matrices in integer and rational powers [13]. Bull. Pol. Ac.: Tech., 45(4), 619‒631 (1997). [10] T. Kaczorek, “Positivity and stability of discrete-time nonlinear systems”, IEEE 2nd International Conference on Cybernetics, 4. Concluding remarks 156‒159 (2015). [11] T. Kaczorek, “Stability of fractional positive nonlinear systems”, The stability of positive linear continuous-time and discrete-time Archives of Control Sciences, 25(4), 491‒496 (2015). systems has been analyzed by the use of the decomposition of [12] T. Kaczorek, Selected Problems of Fractional Systems Theory, the state matrices into symmetrical and antisymmetrical parts. Springer, Berlin 2012. It has been shown that the state Metzler matrix of positive con- [13] T. Kaczorek, “Stability of fractional positive continuous-time tinuous-time linear system is Hurwitz if and only if its sym- linear systems with state matrices in integer and rational pow- A metrical is Hurwitz (Theorem 8). Similarly the state matrix of ers”, Bull. Pol. c.: Tech., 65(3), 305‒311 (2017). [14] T. Kaczorek, “Stability of interval positive fractional dis- positive linear discrete-time system is Schur if and only if its crete-time linear systems:, Int.J.Math.Comput.Sci., 28(3), symmetrical part A I is Hurwitz Theorem 9). These results ¡ 451‒456 (2018). have been extended to inverse matrices of the state matrices of [15] T. Kaczorek, and K. Rogowski, Fractional Linear Systems and the positive linear systems (Theorem 10). A comparison of the Electrical Circuits. Studies in Systems, Decision and Control, stability of positive continuous-time and discrete-time linear vol. 13, Springer 2015. systems has been given (Theorems 11, 12). The considerations [16] W. Mitkowski, “Remarks on stability of positive linear systems”, have been illustrated by numerical examples of positive linear Control and Cybernetics, 29(1), 295‒304 (2000). systems. The considerations can be extended to positive frac- [17] W. Mitkowski, “Dynamical properties of Metzler systems”, Bull. tional linear systems. Pol. Ac.: Tech. 56(1), 309‒314 (2008). [18] M.D. Ortigueira, Fractional Calculus for Scientists and Engi- neers, Springer 2011. Acknowledgment. The studies have been carried out in the [19] P. Ostalczyk, Discrete fractional calculus, World Science Publ. framework of work No. S/WE/1/2016 and financed from the Co., New Jersey 2016. funds for science by the Polish Ministry of Science and Higher [20] I. Podlubny, Fractional Differential Equations. Academic Press: Education. San Diego 1999. I wish to extend my gratitude to D.Sc Lukasz Sajewski for his [21] Ł. Sajewski, “Descriptor fractional discrete-time linear system help in preparation of the final version of this paper. with two different fractional orders and its solution”, Bull. Pol. Ac.: Tech. 64(1), 15‒20 (2016). [22] M.A. Rami and D. Napp, “Characterization and Stability of References Autonomous Positive Descriptor Systems”, IEEE Transactions on Automatic Control, 57(10), 2668‒2673 (2012). [1] A. Berman and R.J. Plemmons, Nonnegative Matrices in the [23] E. Virnik, “Stability analysis of positive descriptor systems”, Mathematical Sciences, SIAM, 1994. Linear Algebra Appl. 429, 2640‒2659 (2008).

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