Positive Switched 2D Linear Systems Described ... - ScienceDirect.com

European Journal of Control (2012)3:239–246 © 2012 EUCA DOI:10.3166/EJC.18.239–246

Positive Switched 2D Linear Systems Described by the Roesser Modelsg Tadeusz Kaczorek Bialystok University of Technology, Faculty of Electrical Engineering, Wiejska 45D, 15-351 Bialystok

The positive switched 2D linear systems described by the Roesser models are addressed. Necessary and sufficient conditions for the asymptotic stability of the positive switched system are established for any switching. The considerations are illustrated by numerical examples. Keywords: Positive, switched, linear, system, Roesser model, stability.

1. Introduction

is given in [1, 7, 11] and some recent result in positive systems has been given in [8–10, 13, 15]. The stability of switched linear systems has been investigated in many papers and books [3, 16, 17]. In this paper the positive switched 2D linear system described by the Roesser models will be considered. We shall analyzed the following question: When is a positive switched 2D linear system defined by a linear Roesser models and a rule describing the switching between them asymptotically stable. It is well known [16, 17] that a necessary and sufficient conditions for stability under arbitrary switching is the existence of a common Lyapunov function for the family of subsystems. This result will be extended for positive switched 2D linear systems described by the Roesser models. The paper is organized as follows. Preliminaries and the problem formulation are given in Section 2. The main results of the paper are presented in Section 3. Necessary and sufficient conditions are established for the asymptotic stability of the positive switched systems for any switching. Illustrating examples are presented in Section 4. Concluding remarks are given in Section 5. To the best the author’s knowledge the positive switched 2D linear systems have not been considered yet.

In positive systems inputs, state variables and outputs take only non-negative values. Examples of positive systems are industrial processes involving chemical reactors, heat exchangers and distillation columns, storage systems, compartmental systems, water and atmospheric pollution models. A variety of models having positive linear behavior can be found in engineering, management science, economics, social sciences, biology and medicine, etc. Positive linear systems are defined on cones and not on linear spaces. Therefore, the theory of positive systems is more complicated and less advanced. An overview of state of the art in positive systems theory is given in the monographs [4, 12]. The most popular models of two-dimensional (2D) linear systems are the discrete models introduced by Roesser [19], Fornasini-Marchesini [5, 6] and Kurek [18]. These models have been extended for positive systems in [12, 14, 20]. An overview of positive 2D systems theory

Let n×m be the set of n × m real matrices. The set n × m matrices with nonnegative entries will be denoted by n×m +

Correspondence to: T. Kaczorek, E-mail: [email protected]

Received 13 July 2010; Accepted 12 May 2011 Recommended by G. Chesi, D. Normand-Cyrot

2. Preliminaries and Problem Formulation

240

T. Kaczorek

n×m (vector x) and n+ = n×1 + . A matrix A = [aij ] ∈ is called strictly positive and denoted by A > 0 (x > 0) if aij > 0 for i = 1, . . ., n; j = 1, . . ., m. The set of nonnegative integers will be denoted by Z+ and the n × n identity matrix will be denoted by In . The Roesser model of 2D linear system has the form [19]

h xi+1,j v xi,j+1

=

A11

A12

A21

A22

yi,j = C1

C2

h xi,j v xi,j

h xi,j

v xi,j

+

B11

B22

ui,j (2.1a)

i, j ∈ Z+

(2.2)

The model (2.1) is called (internally) positive Roesser h ∈ n1 , x v ∈ n2 and y∈ p , i, j ∈ Z model if xi,j + + + i,j + i,j for any nonnegative boundary conditions h x0,j ∈ n+1 , j ∈ Z+ ,

v xi,0 ∈ n+2 , i ∈ Z+

Theorem 2.4: [10] The positive Roesser model is asymptotically stable if and only if one of the following equivalent conditions is met:

(2.3)

|zk | < 1 for k = 1, . . . , n

p(z) = det[In (z + 1) − A] = zn + an−1 zn−1 + · · · + a1 z + a0

A=

A11

are positive, i.e. a¯ 11 a¯ 12 a¯ 11 > 0, > 0, . . . , det A¯ > 0. a¯ 21 a¯ 22

p×n p×m C2 ∈ + , D ∈ + ,

xij = Ak xij ,

n = n1 + n2 . (2.4)

The positive Roesser model (2.1) is called asymptotically stable if for any bounded boundary conditions (2.3) and zero inputs ui,j = 0, i, j ∈ Z+ h v lim xi,j = 0 and lim xi,j = 0.

i,j→∞

i,j→∞

(2.5)

Theorem 2.2: [8] The positive Roesser model (2.1) is asymptotically stable if and only if there exists a strictly positive vector λ ∈ n+ such that [A − In ]λ < 0.

(2.8d)

Consider the switched positive system consisting of q autonomous positive Roesser models of the form

A12 B11 ∈ n×n ∈ n×m + , B= + , A22 B22

A21

C = C1

(2.8b)

are positive, i.e. ai > 0, i = 0, 1, . . . , n − 1. iii) All principal minors of the matrix ⎤ ⎡ a¯ 11 a¯ 12 · · · a¯ 1n ⎥ ⎢a¯ ⎢ 21 a¯ 22 · · · a¯ 2n ⎥ A¯ = In − A = ⎢ . ⎥ (2.8c) . . . .. .. .. ⎦ ⎣ .. a¯ n1 a¯ n2 · · · a¯ nn

Theorem 2.1: [12] The Roesser model is positive if and only if

(2.8a)

ii) All coefficients ai , i = 0, 1, . . . , n − 1 of the characteristic polynomial

and all input sequences ui,j ∈ m + , i, j ∈ Z+ .

(2.7)

is asymptotically stable, where matrix A is given by (2.4).

(2.1b)

h ∈ n1 and x v ∈ n2 are the horizontal and vertiwhere xi,j i,j cal state vectors at the point (i, j) ui,j ∈ m and yi,j ∈ p are the input and output vectors and Akl ∈ nk ×nl , k, l = 1, 2, B11 ∈ n1 ×m , B22 ∈ n2 ×m , C1 ∈ p×n1 , C2 ∈ p×n2 , D ∈ p×m . Boundary conditions for (2.1a) have the form h v x0,j ∈ n1 , j ∈ Z+ and xi,0 ∈ n2 , i ∈ Z+ .

xi+1 = Axi

i) Eigenvalues z1 , . . . , zn of the matrix A have modules less than 1, i.e.

+ Dui,j

Theorem 2.3: [10] The positive Roesser model (2.1) is asymptotically stable if and only if the positive 1D linear system

(2.6)

where xij

=

h xi+1,j

k = 1, . . . , q

h xi,j

(2.9a)

, xij = v , v xi,j+1 xi,j Ak11 Ak12 k = 1, . . . , q. ∈ n×n Ak = + , Ak21 Ak22

(2.9b)

It is assumed that in the point (pi , ti ), pi , ti ∈ Z+ , i = 1, . . . , q the matrix (2.9b) of the Roesser model jump instantaneously from Ai to Aj for some i = j, i, j = 1, . . . , q. The following question arises: when the switched positive Roesser model (2.9) is asymptotically stable for every switching if every positive Roesser model of the set is asymptotically stable.

241

Positive Switched 2D Linear Systems Described by the Roesser Models

3. Problem Solution To simplify the notation it is assumed q = 2. In this case the switched positive system consists of two positive Roesser models xij = A1 xij

(3.1a)

xij

(3.1b)

= A2 xij

where Ak ∈ n×n + , k = 1, 2 and the switching between them occur in the points (p1 , t1 ), . . . , (pk , tk ), (pk+1 , tk+1 ), . . .

(3.2)

satisfying the condition pk+1 ≥ pk ,

V1 (xij ) = λT1 xij and V2 (xij ) = λT2 xij k = 1, 2, . . .

(3.3)

Theorem 3.1: [11] The solution of the autonomous (ui,j = 0, i, j ∈ Z+ ) positive Roesser model (2.1a) with boundary conditions (2.3) is given by h xi,j v xi,j

=

i

k=0

In [8] it was shown that for positive Roesser model (2.1a) as a Lyapunov function a linear form V (xij ) = λT xij can be chosen, where λ ∈ n+ is strictly positive. For the switched positive system consisting of positive Roesser models (3.1) we choose Lyapunov functions in the form

tk+1 ≥ tk and

pk+1 + tk+1 > pk + tk ,

(3.1b) is unstable. For any bounded boundary conditions (2.3) using (3.4), (3.5) and the first Roesser model we may compute boundary conditions for the second Roesser model which will be bounded since the first model is asymptotically stable. In the similar way we may compute the boundary conditions for the first Roesser model but those boundary conditions will be greater than those for the second Roesser model since it is unstable. Therefore, the switched positive Roesser model will be unstable.

Ti−k,j

0 v xk,0

+

j

l=0

Ti,j−l

h x0,l

0

where the strictly positive vectors λ1 and λ2 satisfy the equations λ1 = A1 λ1 + 1n , 1n = [1

(3.4)

where the transition matrix Tij is defined by ⎧ for i = j = 0 In ⎪ ⎨ Tij = T10 Ti−1,j + T01 Ti,j−1 for i, j ≥ 0 (i + j > 0) , ⎪ ⎩ 0 for i < 0 or j < 0 A11 A12 0 0 . (3.5) , T01 = T10 = A21 A22 0 0 Using (3.4), (3.5) and the boundary conditions (2.3) we can compute for the first model (3.1a) the horizontal vector xijh for 0 ≤ i ≤ p1 , 0 ≤ j ≤ t1 and the vertical vector xijv for 0 ≤ i ≤ p1 , 0 ≤ j ≤ t1 but also the boundary conditions v xph1 ,j for t1 ≤ j ≤ t2 and xi,t for p1 ≤ i ≤ p2 . Next we can 1 v h compute the vectors xij , xij and the boundary conditions for the second positive Roesser model (3.1b). Continuing the procedure we can compute the vectors xijh and xijv for i, j ∈ Z+ . Theorem 3.2: The switched positive system (3.1) is asymptotically stable for any switching (3.2) satisfying (3.3) only if both positive models (3.1) are asymptotically stable. Proof: Without loss of generality we may assume that the first model (3.1a) is asymptotically stable and the second

(3.6)

1

λ2 = A2 λ2 + λ1 , . . . 1]T ∈ n+ .

(3.7)

If A1 and A2 are Schur matrices then from (3.7) we have λ1 = [In − A1 ]−1 1n and λ2 = [In − A2 ]−1 λ1 .

(3.8)

Lemma 3.1: The function V2 (xij ) = λT2 xij

(3.9)

is a common Lyapunov function for the both positive Roesser models (3.1) if A1 A2 = A2 A1 .

(3.10)

Proof: The function (3.9) for both positive Roesser models (3.1) for strictly positive λ2 ∈ n+ is positive if and only if xij = 0. Note that the dual Roesser models xij = AT1 xij

(3.11a)

xij = AT2 xij

(3.11b)

are positive and asymptotically stable if and only if the corresponding Roesser models (3.1) are positive and asymptotically stable [8]. Using (3.9) for the positive Roesser model (3.1a) we obtain V2 (xij ) = λT2 [A1 − In ]xij

(3.12)

λT2 = λT1 [In − A2 ]−1

(3.13)

and

since λ2 = AT2 λ2 + λ1 .

242

T. Kaczorek

Figure 1a. State vector of the first system with A1 .

Figure 1b. State vector for second system with A2 .

is asymptotically stable for any switching (3.2) satisfying (3.3) if and only if the positive Roesser models (3.1) are asymptotically stable.

Substitution of (3.13) into (3.12) yields V2 (xij ) = λT1 [In − A2 ]−1 [A1 − In ]xij = λT1 [A1 − In ][In − A2 ]−1 xij = −1Tn [In − A2 ]−1 xij < 0 for every xij ∈ n+ ,

xij = 0

(3.14)

since (3.10) implies [A1 − In ][In − A2 ]−1 = [In − A2 ]−1 [A1 − In ] and the sum of entries of each column of the matrix 1Tn [In − A2 ]−1 is positive for the positive asymptotically stable Roesser model (3.1b) . Similarly, using (3.9) for positive Roesser model (3.11b) we obtain V2 (xij ) = λT2 [AT2 − In ]xij < 0 for every xij ∈ n+ , since λT2 [AT2 − In ] < [0

xij = 0

(3.15)

. . . 0] see [8].

Proof: Necessity follows immediately from Theorem 3.2. If the condition (3.10) is met and the Roesser models (3.1) are asymptotically stable then by Lemma 3.1 the function (3.9) is a common Lyapunov function for the positive models (3.1) which satisfies the conditions (3.14) and (3.15). Therefore, the positive switched system (3.1) is asymptotically stable.

4. Illustrating Examples Example 4.1: Consider the positive switched system consisting of two Roesser models (3.1) with the matrices 2 1 0.1 0 A1 = , A2 = . (4.1) 0 0.4 1 0.2 The switching occurs between them in the points

Theorem 3.3: Let the matrices A1 and A2 of (3.1) satisfy the conditions (3.10). The positive switched system (3.1)

(2, 1), (3, 3), (5, 4), (6, 6), . . .

(4.2)

243


Fig. 2. State vector of the system with switches.

Fig. 3. State vector of the system with switches.

h = x v = 1, x h = x v = 0 For boundary conditions: x00 00 i0 0j and i, j = 1, 2, . . . we have The first Roesser model with A1 is unstable and the second one with A2 is asymptotically stable (Fig. 1a) and (Fig. 1b). By Theorem 3.2 the positive switched system is unstable since lim xijh = ∞ (Fig. 2). j→∞

Example 4.2: Consider the positive switched system consisting of two Roesser models (3.1) with the matrices 0.4 A1 = 0.3

0.1 , 0.2

0.6 A2 = 0.6

0.2 . 0.2

(4.3)

The switching occur between them in the points (4.2). In this case both Roesser models are asymptotically stable and the matrices satisfy the condition (3.10). By Theorem 3.3 the positive switched system with (4.3) and (4.2) is asymptotically stable (Fig. 3).

Example 4.3: Consider the positive switched systems consisting of two Roesser models (3.1) with the matrices Ak11 Ak12 , k = 1, 2 (4.4a) Ak = Ak21 Ak22 where

A111

a = 0.2

0.1 , 0.3

A112

0.2 = , 0

A121 = [0.1 0.2], A122 = [0.4], 0.4 0.3 0.1 2 2 A11 = , A12 = , 0.2 0.5 0.2 A221 = [0

0.1],

A222 = [0.3].

(4.4b)

The switching occur between them in the points (4.2).

244

T. Kaczorek



Figure 4c. State vector of the system with switches.

The matrices (4.4) do not satisfies the condition (3.10) for any values of the coefficient a. The positive switched system with (4.4) and (4.2) is asymptotically stable for a < 1 (Fig. 4) and it is unstable for a > 1 (Fig. 5). The presented considerations can be easily extended to the positive switched linear systems consisting of q (q > 2) autonomous Roesser models.

5. Concluding Remarks The positive switched 2D linear systems described by the Roesser models have been addressed. Necessary and sufficient conditions have been established for the asymptotic stability of the positive switched systems for any switching.

245




Figure 5c. State vector of the system with switches.

The considerations can be easily extended for positive switched 2D linear systems described by the FornasiniMarchesini models and the general 2D model.

Acknowledgment This work was supported by Ministry of Science and Higher Education in Poland under work S/WE/1/11.

References 1. Bose NK. Multidimensional Systems Theory Progress, Directions and Open Problems, D. Reidel Publishing Co., 1985. 2. Buslowicz M. Simple stability conditions for linear discretetime systems with delays. Bull. Pol. Acad. Sci., 56(4): 325– 328. 3. Colaneri P. Dwell time analysis of deterministic and stochastic switched systems. Eur. J. Control, 2009; 15(3–4): 228–248.

246 4. Farina L., Rinaldi S. Positive Linear Systems, Theory and Applications. John Wiley, New York, 2000. 5. Fornasini E., Marchesini G. State-space realization theory of two-dimensional filters. IEEE Trans., Autom. Contr., 1976; AC-21: 481–491. 6. Fornasini E., Marchesini G. Double indexed dynamical systems. Math. Sys. Theory, 1978; 12: 59–72. 7. Gałkowski K. State Space Realizations of Linear 2D Systems with Extensions to the General nD (n > 2) Case, SpringerVerlag, London, 2001. 8. Kaczorek T. Choice of the forms of Lyapunov functions for positive 2D Roesser model. Int. J. Appl. Math. Comput. Sci., 2007; 17(4): 471–475. 9. Kaczorek T. Independence of the asymptotic stability of positive 2D linear systems with delays of their delays. Int. J. Appl. Match. Comput. Sci., 2009; 19(2): 255–261. 10. Kaczorek T. Asymptotic stability of positive 2D linear systems with delays. Bull. Pol. Acad. Sci. Techn., 2009; 27(2): 133–138. 11. Kaczorek T. Two-dimensional Linear Systems, Springer Verlag, Berlin, 1985. 12. Kaczorek T. Positive 1D and 2D Systems, Springer-Verlag, London, 2001.

T. Kaczorek

13. Kaczorek T. When do equlibria of the positive 2D Roesser model are strictly positive. Bull. Acad. Pol. Sci. Techn., 2002; 50(1): 221–227. 14. Kaczorek T. Reachability and controllability of non-negative 2D Roesser type models. Bull. Acad. Pol. Sci. Techn., 1996; 44(4): 405–410. 15. Kaczorek T. Reachability and minimum energy control of positive 2D systems with delays. Control Cybernetics, 2005; 34(2): 411–423. 16. Liberzon D. Switching in Systems and Control, Birkhoiser, Boston, 2003. 17. Liberzon D. On new sufficient conditions for stability of switched linear systems. Proc. European Control Conf. 2009, Budapest, August 23–26, pp. 3257–3262. 18. Kurek J. The general state-space model for a twodimensional linear digital systems. IEEE Trans. Autom. Contr., 1985; AC-30: 600–602. 19. Roesser RP. A discrete state-space model for linear image processing. IEEE Trans. Autom. Contr., 1975; AC-20(1): 1–10. 20. Valcher ME. On the internal stability and asymptotic behavior of 2D positive systems. IEEE Trans. Circuits Systems—I, 1997; 44(7): 602–613.