dr inĪ. àukasz Sajewski
Bialystok University of Technology Faculty of Electrical Engineering
DODATNIE LINIOWE UKàADY CIĄGàO-DYSKRETNE
Z OPÓħNIENIEM W WEKTORZE STANU
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POSITIVE CONTINUOUS-DISCRETE TIME LINEAR SYSTEMS WITH DELAYS IN STATE VECTOR
A new class of positive continuous-discrete time linear systems with delays in state vector is addressed. Three state space model of this class of linear systems are considered. Necessary and sufficient conditions for the positivity of continuous-discrete time systems with delays in state vector are established. The proper transfer matrix of this class of linear systems is given.
1. INTRODUCTION
In positive system s inputs, state variables and outputs take only non-negative values.
Examples of positive system s are industrial p rocesses in volving ch emical reactors, heat exchangers and distillation colum ns, storage s ystems, compartm ental system s, water an d atmospheric pollution models. A va riety of models having posit ive linear system s behavior can be found in engineering, m anagement science, econom ics, social sciences, biology and medicine, etc. Positiv e lin ear system s are defined on cones and not on linear spaces. Therefore, the theo ry of positiv e system s is m ore com plicated and less advanced. An overview of state of the art in positive systems is given in the monographs [6, 9].
2D hybrid system is dynamic systems that incorporate both continuous-time and discrete-time dynamics. It means that state vector, input and output vectors of 2D hybrid system depend on the continuous time t and the discrete variable i. Examples of hybrid systems include systems with relays, switches an d hysteresis, transmissions, and other m otion controllers, constrained robotic system s, automated highway system s, flight control, m anagement system s and analog/digital circuit. The pos itive continuous-discrete 2D linear system s have been introduced in [8], positive hybrid linear system s in [10] and the positiv e fractional 2D hybrid systems in [11]. Different m ethods of solvability of 2D hybrid linear system s ha ve been discussed in [13] and the solution to singular 2D hybrids linear system s has been derived in [15]. The realization problem for positiv e 2D hybrid system s has been addressed in [12]. Some problems of dynamics and control of 2D hybrid systems have been considered in [5, 7]. The problems of stability and r obust stability of 2D continuous-discrete linear system s have been investigated in [1-4]. Recently the stability and robust stability of Fornasini-Marchesini type model and of Roesser type m odel of scalar continuous-discrete linear systems have been analyzed by Buslowicz in [2–4].
The m ain goal of this paper is to present a new class of positiv e con tinuous-discrete tim e linear systems with delays in state vector. Necessary and sufficient conditions for positivity of this class of linear systems will be established. Three state space model called: general model, first Fornasini-March esini m odel and seco nd Fornasini-Marchesini m odel of linear continuous-discrete time systems with delays w ill be considered. The proper transfer m atrix of this class of linear systems will be given.
The paper is organized as follows. In subsection 2.1 the general m odel of the continuous-discrete time linear system s with delays and its boundary conditions are presented. In this subsection the definition and conditions for positivity of this class of linear system s are given. Subsections 2.2 and 2.3 contains the sam e considerations as in 2.1 for first F ornasini-Marchesini model and second Fornasini-ornasini-Marchesini m odel respectively. In subsection 2.4 results according to transfer m atrix of the continuous-discrete tim e systems with delays are presented. Concluding remarks are given in section 3.
The following notation will be used: - the set of real numbers, Z – the set of nonnegative integers, num – the s et of
m
nu rea l m atrices, num – the set of num m atrices with
nonnegative entries and u1
n n , M
n – the set of nun Metzler matrices (real matrices with
nonnegative off-diagonal entries), I – the n nun identity matrix.
2. THREE MODELS OF POSITIVE CONTINUOUS-DISCRETE TIME SYSTEMS WITH DELAYS
2.1 General model of the continuous-discrete time system with delays
Consider a hybrid system with delays described by the equations ) 1 , ( ) , ( ) , ( ) 1 , ( ) 1 , ( ) 1 , ( ) 1 , ( ) , ( ) , ( ) 1 , ( 2 1 0 5 4 3 2 1 0 i t u B i t u B i t u B i d t x A i t x A i d t x A i t x A i t x A i t x A i t x (2.1a) ) , ( ) , ( ) , (t i Cx t i Du t i y , t [ f0, ], iZ {01,,...} (2.1b) where t i t x i t x w w ( , ) ) , ( , n i t x( , ) , u(t,i)m, y(t,i)p and Aknun, k 01,,2,3,4; , m n
B u Cpun, Dpum are the real matrices, d !0 is a delay.
Boundary conditions for (1a) have the form ) , ( 0 t i xi , ]t[ d ,0 , i Z, )xt0(t,i , )xt0(t,i , ]i[ ,10 , t (2.2a) and 0 ) 0 , ( 0 t x i , ]t[ d ,0 , 0xt0(0,i) xt0(0,i) , ]i[1,0 (2.2b)
Definition 2.1. The hybrid sys tem with delays (2.1) is called (internally ) p ositive if
n
i t
x( , ) nd y(t,i)p, ta, iZ for arbitrary boundary conditions x0i(t,i)n, ]
0 , [ d
t , iZ i xt0(t,i)n, xt0(t,i)n, ]i[ ,10 , t and all inputs u(t,i)m,
t , iZ.
Theorem 2.1. The hybrid system with delays (2.1) is internally positive if and only if
m p n p m n n n n n n D C B B B A A A A A A A A M A u u u u u , , , , , , , , , , 2 1 0 2 1 0 5 4 3 1 0 2 (2.3) Proof.
Necessity. Let ei be the i-th ( i 1,...,n) column of the identity m atrix In. From (2.1) for t , i = 0 and x(t1,) ei, x(t,0)0 , 0 x(t,0) , 0x(t d1,) , 0 x(t,1) , 0 ) 1 , (t d
x and inputs u(t,0) u(t,0) u(t1,) 0 we have x(t,1) A2ei. The trajectory does not liv e the o rthant n
R only if A2ei t0, what im plies aij t0, iz j. Therefore, the matrix A2 has to be th e Metzler m atrix. For the sam e reasons for x(01,) 0, x(0,0) 0,
0 ) 0 , 0 ( x , x(d1,) 0, x(0,1) 0, x(t d,1) 0, n u B u B u B
m
R u
u
u(0,0), (0,0), (01,) m ay be arbitrary. Sim ilarly, for t R, i = 0 and x(t1,) 0, 0 ) 0 , (t x , x(t d1,) 0, x(t,1) 0, x(t d,1) 0 we have n t x A t x( ,1) 1( ,0) what implies n n
A1u since x(t,0)n may be arbitrary. F or the sam e reasons for x(01,) 0, 0 ) 0 , 0 ( x , x(d1,) 0, x(0,1) 0, x(t d,1) 0, x(01,) A0x(0,0)n, what im plies n n
A0u since x(0,0)n may be arbitrary. F rom (1b) for u(0,0) 0 we have
p
Cx
y(0,0) (0,0), what im plies Cpun, since x(0,0)n may be arbitrary. For the same reasons for x(0,0) 0 we have y(0,0) Du(0,0)p, what implies Dpum, since
m
u(0,0) may be arbitrary.
In a similar way we m ay show that the hybrid s ystem with delays (2.1 ) is internally positive only if the conditions (2.2) are satisfied.
Sufficiency. From (2.1) for i = 0 and t[ d0, ]we have
) 0 , ( ) 1, ( ) 1, ( ) 1, (t A2x t A3x t d F t x (2.4a) where ) 1 , ( ) 0 , ( ) 0 , ( ) 1 , ( ) 1 , ( ) 0 , ( ) 0 , ( ) 0 , (t A0x t A1x t A4x t A5x t d B0u t B1u t B2u t F (2.4b)
For given nonnegative initial conditions
) 1, 0 ( ) 1, ( ), 1 , ( ), 1 , ( ), 0 , ( ), 0 , (t x t x t x t d x t d x0i x | and u(t,0),u(t,0),u(t1,)m, ] , 0 [ d t , we obtain F(t,0)n, t[ d0, ] if A0,A1,A4,A5nun, B0,B1,B2num. he T solution of the equation (2.4a) has the form
W
W
W
W d F d x A e x e t x t t A t A³
0 3 ) ( ( 1,) ( ,0) ) 1, 0 ( ) 1, ( 2 2 (2.5)and is nonnegative since At n n
e 2 u for t[ d0, ] if and only if
2
A is the Metzler matrix and
n n
A3u . Knowing x(t,1) for t[ d0, ] in a s imilar way we can find x(t,1) for t[d,2d], ] 3 , 2 [ d d t ,… .
From (2.1a) for i = 1 we have
) 1 , ( ) 2 , ( ) 2 , ( ) 2 , (t A2x t A3x t d F t x (2.6a) where ) 2 , ( ) 1, ( ) 1, ( ) 0 , ( ) 0 , ( ) 1 , ( ) 1, ( ) 1, (t A0x t A1x t A4x t A5x t d B0u t B1u t B2u t F (2.6b)
Substituting (2.5) into (2.6b) we obtain
( 1,) ( ,0)
( ,0) ( ,0) ( 1,) ( 1,) ( ,2) ) 0 , ( ) 1, ( ) 1, ( ) 0 , 0 ( ) 1, 0 ( ) ( ) 2 , ( ) 1, ( ) 1, ( ) 0 , ( ) 0 , ( ) 0 , ( ) 1, ( ) 1, 0 ( ) 0 , ( ) 1, ( ) 1, 0 ( ) 1, ( 2 1 0 5 4 3 1 3 0 3 2 1 0 2 1 0 5 4 0 ) ( 0 3 ) ( 1 0 ) ( 0 0 3 ) ( 0 0 2 2 2 2 2 2 2 t u B t u B t u B d t A t x A t F d t x A A t F d t x A A d x A F x A A A e t u B t u B t u B d t A t x A d F e d d x A e x e dt d A d F e A d d x A e A x e A t F t A t t A t t A t A t t A t t A t A ¸¸ ¹ · ¨¨ © §
³
³
³
³
W
W
W
W
W
W
W
W
W W W W (2.6c)For given nonnegative initial conditions (2.2), nonnegative inputs we obtain n
t F( 1,), f i and only if nn n n nm B B B A A A A A A
A0, 1, 4, 5u , 0 1 2u , 0, 1, 2u . The solu tion of the equation (2.6a) has the form
n t t A t A d F d x A e x e t x
³
W W W W 0 3 ) ( ( ,2) ( ,1) ) 2 , 0 ( ) 2 , ( 2 2 (2.7)if the conditions (2.3) are met.
Continuing the procedure for i !0 with nonnegative initial conditions (2.2), nonnegative inputs and conditions (3) we may show that
n i t u B i t u B i t u B i d t x A i t x A i t x A i t x A i t F( , ) 0 ( , ) 1( , ) 4( , 1) 5 ( , 1) 0 ( , ) 1( , ) 2 ( , 1) (2.8a) and
n t t A t A d i F i d x A e i x e i t x
³
W W W W 0 3 ) ( ( , 1) ( , ) ) 1 , 0 ( ) 1 , ( 2 2 (2.8b) for tand i Z. Ƒ2.2. First Fornasini-Marchesini model of continuous-discrete time system with delays Let consider a general m odel (2.1) with B B0, B1 B2 0. We obtain so called first Fornaisni-Marchesini model of the continuous-discrete time system with delays in state vector described by the equations
) , ( ) 1 , ( ) 1 , ( ) 1 , ( ) 1 , ( ) , ( ) , ( ) 1 , ( 5 4 3 2 1 0 i t Bu i d t x A i t x A i d t x A i t x A i t x A i t x A i t x (2.9a) ) , ( ) , ( ) , (t i Cx t i Du t i y , t [ f0, ], iZ {01,,...} (2.9b) where t i t x i t x w w ( , ) ) , ( , n i t x( , ) , u(t,i)m, y(t,i)p and Aknun, k 01,,2,3,4,5; , m n
B u Cpun, Dpum are the real matrices, d !0 is a delay.
Boundary conditions for (9a) have the form (2.2). The necessary and sufficient conditions for the positivity of this model is given by the following theorem.
Theorem 2.2. The continuous-discrete tim e system with delays (2.9) is internally p ositive if and only if m p n p m n n n n n n D C B A A A A A A A M A u u u u u , , , , , , , 0 1 3 4 0 1 2 2 (2.10)
The proof is similar as in Theorem 2.1.
2.3. Second Fornasini-Marchesini model of continuous-discrete time system with delays Let consider a general model (2.1) with A0 0 and B0 0. W e obtain so called second Fornaisni-Marchesini model of the continuous-discrete time system with delays in state vector described by the equations
) 1 , ( ) , ( ) 1 , ( ) 1 , ( ) 1 , ( ) , ( ) 1 , (t i A1x t i A2x t i A3x td i A4x t i B1u t i B2u t i x (2.11a) ) , ( ) , ( ) , (t i Cx t i Du t i y , t [ f0, ], iZ {01,,...} (2.11b) where t i t x i t x w w ( , ) ) , ( , n i t x( , ) , u(t,i)m, y(t,i)p and Ak nun, k ,12,3,4; , , 2 1 B nm
B u Cpun, Dpum are the real matrices, d !0 is a delay.
Boundary conditions for (1a) have the form (2.2). The necessary and sufficient conditions for the positivity of this model is given by the following theorem.
Theorem 2.3. The hybrid system with delays (2.1) is internally positive if and only if
m p n p m n n n n n n D C B B A A A A A M A u u u u u , , , , , , , 2 1 2 1 4 3 1 2 (2.12)
The proof is similar as in Theorem 2.1.
2.4. Transfer matrix of the continuous-discrete time system with delays
Let consider the general m odel of the continuous -discrete time system with delays in state vector (2.1). Using the Laplace tran sform )(L with zero boundary conditio ns according to t we obtain ) , ( ) , ( ) , ( ) 1 , ( ) , ( ) , ( ) 1 , ( ) 1 , ( ) 1 , ( ) 1 , ( ) , ( ) , ( ) 1 , ( 2 1 0 5 4 3 2 1 0 i s DU i s CX i s Y i s U B i s sU B i s U B i s X e A i s sX A i s X e A i s X A i s sX A i s X A i s sX sd sd (2.13a) Then using Z (Z transform with zero boundary conditions according to i we obtain )
) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( ) , ( 2 1 0 0 1 5 1 4 3 2 1 0 z s DU z s CX z s Y z s zU B z s sU B z s U B i s U B z s X z e A z s sX z A z s zX e A z s zX A z s sX A z s X A z s szX sd sd (2.13b) Substituting z1 wd , wc esd we have ) , ( ) , ( ) , ( ) , ( ) ( ] ) ( ) ( ) ( [ ) , ( 1 0 1 2 3 2 4 1 5 0 z s DU z s CX z s Y z s U z B s B B z w A A s w A A w w A A sz I C z s X n c d d c (2.12) where Xk Xk(s,z) Z
^
L[xk(t,i)]`
, k = 1,2; U U(s,z) Z^
L[u(t,i)]`
,^
[ ( , )]`
) , (s z y t i Y Y Z L .From (2.12) we have the following theorem.
Theorem 2.4. Transfer matrix of the system (2.1) is given by equation
D z B s B B z w A A s w A A w w A A sz I C D z s T z s U z s Y z s T c d d c n sp ( ) ] ) ( ) ( ) ( [ ) , ( ) , ( ) , ( ) , ( 2 1 0 1 3 2 4 1 5 0 (2.13) where )Tsp( zs, is the strictly proper transfer matrix, wc esd, wd z1 is the representation of delay for continuous-time and discrete-time part of the system (2.1) respectively.
Substituting B B0, B1 B2 0 or A0 0, 0B0 into (2.13) we o btain transfer m atrix of the first F ornasini-Marchesini model and second Fornasini-Marchesini m odel of the continuous-discrete time system with delays respectively.
For m input and p output continuous-discrete tim e system with delay s in state v ector the proper transfer matrix takes the form
) , ( ) , ( ... ) , ( ... ) , ( ... ) , ( ) , ( , 1 , , 1 11 z s z s T z s T z s T z s T z s T pm m p p m u » » » ¼ º « « « ¬ ª # # (2.14) where ) ( ) ( ) ( ... ) ( ) ( ) ( ) ( ) ( ... ) ( ) ( ) , ( , 00 , 01 , 10 1 , , 1 1 , 1 , , 00 , 01 , 10 1 , , 1 1 , 1 , , , , w a z w a s w a z s w a z s w a z s w b z w b s w b z s w b z s w b z s b z s T kl n n kl kl kl n n n n l k n n n n l k l k l k n n l k n n n n l k n n n n l k n n l k (2.15) for k ,12,...,p; l ,12,...,m; and
¦¦
¦¦
p n i q n j j d i c kl pq j i l k q p p n i q n j j d i c kl pq j i l k q p w w a w a w w b w b 0 0 , , , , 0 0 , , , , ) ( ) ( (2.16)for p, q 01,,...,n, ,, ( ) ,, , ,, ( ) 1 2 1 w a b w
bnknl nknl nkln and pum( zs, ) is the se t o f mun rational matrices in s and z.
Definition 2.2. The m atrices (2.3) are called the positiv e realization of the transfer m atrix )
, ( zs
T if they satisfy the equality (2.13). 3. CONCLUDING REMARKS
The new class of positive continuous-discrete tim e linear systems with delays in state vector has been in troduced. Necessary and sufficient conditions fo r the positiv ity of this class of linear system s with delays has been established. Three model of continuous-discrete time linear systems with delays in state vector have been proposed. The proper transfer m atrix of this class o f linear system s has be en given. The considerations can be also extended for fractional positive 2D continuous-discrete linear systems.
ACKNOWLEGMENT
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