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 nonlin- ear 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 n lim x = 0 for all x ℜ . (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 det I λ (– A) = ¡ = λ2 + 4λ + 3 (20) ̂ ̂ ¡ 11 12 2 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.