EXTERNALLY AND INTERNALLY POSITIVE TIME-VARYING LINEAR SYSTEMS

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1 Int. J. Appl. Math. Comput. Sci., 1, Vol.11, No.4, EXTERNALLY AND INTERNALLY POSITIVE TIME-VARYING LINEAR SYSTEMS Tadeusz KACZOREK The notions of externally and internally positive time-varying linear systems are introduced. Necessary and sufficient conditions for the external and internal positivities of time-varying linear systems are established. Moreover, sufficient conditions for the reachability of internally positive time-varying linear systems are presented. 1. Introduction Roughly speaking, positive systems are systems whose trajectories are entirely contained in the nonnegative orthant + whenever the initial state and input are nonneg- n ative. Positive systems arise while modelling systems in engineering, economics, social sciences, biology, medicine and other areas (d Alessandro and de Santis, 1994; Berman et al., 1989; Berman and Plemmons, 1994; Farina and Rinaldi, ; Kaczorek, 1; Rumchev and James, 199; 1995). The single-input single-output externally and internally positive linear time-invariant systems were investigated in (Berman et al., 1989; Berman and Plemmons, 1994; Farina and Rinaldi, ). The notions of externally and internally positive systems were extended to singular continuous-time, discrete-time and two-dimensional linear systems in (Kaczorek, 1). The reachability and controllability of standard and singular internally positive linear systems were analysed in (Fanti et al., 199; Klamka, 1998; Otha et al., 1984; Valcher, 1996). The notions of weakly positive discrete- and continuous-time linear systems were introduced in (Kaczorek, 1998b; 1). Recently, the positive two-dimensional (D) linear systems have been extensively investigated by Fornasini and Valcher (Fornasini and Valcher, 1997; Valcher, 1996; 1997) and (Kaczorek 1). Institute of Control and Industrial Electronics, Faculty of Electrical Engineering, Warsaw University of Technology, ul. Koszykowa 75, 66 Warszawa, Poland, kaczorek@isep.pw.edu.pl

2 958 T. Kaczorek. Preliminaries p q Let be the set of real p q matrices and p := p 1. The set of p q p q p p 1 real matrices with nonnegative entries will be denoted by + and + := Consider the linear time-varying system +. ẋ(t) = A(t)x(t) + B(t)u(t), x( ) = x, (1a) y(t) = C(t)x(t) + D(t)u(t), (1b) where ẋ(t) = dx(t)/dt, x(t) n is the state vector, u(t) m signifies the input vector, y(t) p stands for the output vector, and A(t), B(t), C(t), D(t) are real matrices of appropriate dimensions with continuous-time entries. A solution x(t) to the equation satisfying the initial condition x( ) = x is given by (Gantmacher, 1959) t x(t) = Φ(t, )x + where Φ(t, ) is the fundamental matrix defined by t t Φ(t, ) = I n + A(τ) dτ + Φ(t, τ)b(t)u(τ) dτ, () τ A(τ) A(τ 1 ) dτ 1 dτ +, (3) I n being the n n identity matrix. If A(t 1 )A(t ) = A(t )A(t 1 ) for t 1, t [, ), then (3) takes the form (Gantmacher, 1959) ( t Φ(t, ) = exp ) A(τ) dτ. (3a) The fundamental matrix Φ(t, ) satisfies the matrix differential equation Φ(t, ) = A(t)Φ(t, ), (4) and the initial condition Φ(, ) = I n. 3. Externally Positive Systems Definition 1. The system (1) is called externally positive if for all u(t) m +, t and zero initial conditions (x = ) the output vector y(t) p + for t. Let g(t) p m be the matrix impulse response of the system (1). It is wellknown that the output vector y(t) of the system (1) with zero initial conditions for an input vector u(t) is given by the formula y(t) = t g(t, τ)u(τ) dτ, t, (5)

3 Externally and internally positive time-varying linear systems 959 where g(t, τ) = C(t)Φ(t, τ)b(t) + D(t)δ(t τ), (6) for t τ, and δ(t) is the Dirac impulse. Theorem 1. The system (1) is externally positive if and only if g(t) p m + for t. (7) Proof. The necessity follows immediately from Definition 1 and the definition of the impulse response. To show the sufficiency, assume that (7) holds. Then from (5), for u(t) m +, t we have y(t) p + for t. 4. Internally Positive Systems Definition. System (1) is called internally positive if for every x n + and all u(t) m + the state vector x(t) n + and y(t) p + for t. From the comparison of Definitions 1 and it follows that every internally positive system (1) is always externally positive. Lemma 1. The fundamental matrix satisfies Φ(t, ) n n + for t, (8) if and only if the off-diagonal entries a ij, i j, i, j = 1,..., n of the matrix A(t) satisfy the condition t a ij (τ) dτ for i j, i, j = 1,..., n. (9) Proof. First, we shall show that (9) implies (8). Let x i (t) (resp. z i (t)) be the i-th component of the vector x(t) (resp. z(t)) and ( t x i (t) = z i (t) exp ) a ii (τ) dτ, i = 1,..., n. (1) Substitution of (1) into (1a) for u(t) =, t yields (Ratajczak, 1967) ż(t) = Ā(t)z(t), (11) where Ā(t) = [ā ij(t)] n n ā ij (t) = ( t a ij (t) exp [ ajj (τ) a ii (τ) ] ) dτ for i j, for i = j, (1)

4 96 T. Kaczorek From (1) it follows that z i ( ) = x i ( ) for i = 1,..., n if x n +. (13) Using () for u(t) =, t and (3) for (11), we obtain z(t) = Φ(t, )z, (14) where t t Φ(t, ) = I n + Ā(τ) dτ + τ Ā(τ) Ā(τ 1 ) dτ 1 dτ +. (15) From (1) it follows that if (9) holds, then Ā(t), and by (15) this implies Φ(t, ) + n n and z(t) n +, t for any z n +. Hence, by (1) and (13) we have x(t) n +, t for any x n +. Therefore (9) implies (8). The necessity follows immediately from (3a) and the fact that Φ(t, ) + n n only if t Ā(τ) dτ is a Metzler matrix for any t (Kaczorek, 1998a). n n + Remark 1. If the matrix A(t) is independent of t, i.e. A(t) = A = [a ij ] and a ij for i j, then A is the Metzler matrix (Farina and Rinaldi, ; Kaczorek, 1) and Φ(t, ) = exp(a(t )). Theorem. System (1) is internally positive if and only if (i) the off-diagonal entries of A(t) satisfy (9), (ii) B(t) + n m, C(t) p n +, D(t) p m + for t. Proof. (Necessity) Let u(t) = for t and x = e j. The trajectory does not n leave the orthant + only if ẋ( ) = A( )e j, and this implies (9). For the same reasons, for x = we have ẋ( ) = Bu( ), and this implies B(t) p m, t since u( ) m + may be arbitrary. From (1b), for u( ) = we have y( ) = C( )x p p n + and C(t) +, t since x n + may be arbitrary. Similarly, from (1b), for x = we obtain y( ) = D( )u( ) p p m + and D(t) + for t since u( ) m + may be arbitrary. (Sufficiency) If the condition (9) is satisfied, then, by Lemma, (8) holds and from () we obtain x(t) n + for any x n + and u(t) m +, t, since B(t) + n m. If C(t) p n + and D(t) + p m for t, then from (1b) we obtain y(t) p + since x(t) n + and u(t) m + for t.

5 Externally and internally positive time-varying linear systems Reachability Definition 3. The state x f (t) n + of the system (1) is called reachable in time t f if there exists an input vector u(t) m + for [, t f ] which steers the state of the system from x = to x f. Definition 4. If every state x f (t) n + of the system (1) is reachable in time t f, then the system is called reachable in time t f. Definition 5. If for every state x f (t) n + there exists t f > such that the state is reachable in time t f, then the system (1) is called reachable. A matrix A + n n is called monomial (or the generalised permutation matrix) if in each row and in each column only one entry is positive and the remaining entries are zero. Theorem 3. The internally positive system (1) is reachable in time t f if R(t f, ) := t f Φ(t f, τ)b(τ)b T (τ)φ T (t f, τ) dτ (T denotes the transpose) (16) is a monomial matrix. The input vector which steers the state vector of (1) from x = to x f is given by for t [, t f ]. u(t) = B T (t)φ T (t f, t)r 1 (t f, t)x f, (17) Proof. If R(t f, ) is a monomial matrix, then R 1 (t f, ) + n n and u(t) m + for [, t f ]. We shall show that (17) steers the state of (1) from x = to x f. Substituting (17) into () for t = t f and x =, we obtain x(t f ) = = t f [ t f Φ(t f, τ)b(τ)b T (τ)φ T (t f, τ)r 1 (t f, )x f dτ ] Φ(t f, τ)b(τ)b T (τ)φ T (t f, τ) dτ R 1 (t f, )x f = x f. Therefore, if (16) is a monomial matrix, then the positive system (1) is reachable in time t f. Theorem 4. The internally positive system (1) is reachable in time t f if A(t) = diag [a 1 (t), a (t),..., a n (t)], (18) (a i (t), i = 1,..., n is continuous-time function) and B(t) n n + is a monomial continuous-time matrix.

6 96 T. Kaczorek Proof. It is well known (Gantmacher, 1959) that if A(t) has the form (18), then A(t 1 )A(t ) = A(t )A(t 1 ) for t 1, t [, ) and Φ(t, ) = exp( t A(τ) dτ) is also a diagonal nonnegative matrix for t. Hence the matrix Φ(t, )B(t) + n n is a monomial matrix and so is the matrix R(t f, ) = t f Φ(t f, τ)b(τ)b T (τ)φ T (t f, τ) dτ = t f Φ(t f, τ)b(τ)[φ(t f, τ)b(τ)] T dτ. Then, by Theorem 3, the system (1) is reachable in time t f. Remark. If the diagonal matrix (18) and B(t) are independent of t, then from Theorems 3 and 4 we obtain the corresponding theorems 3.1 and 3.11 in (Kaczorek, 1). Similar results can be obtained for the controllability of time-varying linear systems. 6. Example Consider system (1) with = and A(t) =, B(t) = et t. (19) By Theorem 4, the system is reachable in time t f. Therefore there exists an input u(t) which steers the state of the system from x = to x f = [ 1] T in time t f = 1. Using (3a), (16) and (17), we obtain ( 1 Φ(1, τ) = exp R(t f, ) = R(1, ) = τ ) A(τ) dτ 1 exp ( (1 τ) ) = ( 1 ( exp )) 1 τ Φ(1, τ)b(τ)b T (τ)φ T (1, τ) dτ = u(t) = B T (t)φ T (1, t)r 1 (1, )x f = 4 exp( t) e 1, e 4 ( ) 1 e t e 1 exp 1 (e 1) ( 1 ( )) 1 t.,

7 Externally and internally positive time-varying linear systems Concluding Remarks The notions of externally and internally positive time-varying linear systems were introduced. Necessary and sufficient conditions for the external and internal positivities of time-varying linear systems were established. The concept of reachability was extended to internally positive time-varying linear systems, and sufficient conditions for the reachability of internally positive time-varying linear systems were established. With minor modifications, the consideration can be extended to discrete time-varying linear systems. A generalization to D linear systems with variable coefficients is also possible. An open problem is an extension of the consideration to singular time-varying linear systems. References d Alessandro P. and de Santis E. (1994): Positiveness of dynamic systems with non-positive coefficients matrices. IEEE Trans. Automat. Contr., Vol.AC-39, pp Berrman A., Neumann M. and Stern R. (1989): Nonnegative Matrices in Dynamic Systems. New York: Wiley. Berman A. and Plemmons R.J. (1994): Nonnegative Matrices in Mathematical Sciences. Philadelphia: SIAM Press. Fanti M.P., Maione B. and Turchsano B. (199): Controllability of multi-input positive discrete-time systems. Int. J. Contr., Vol.51, No.6, pp Farina L. and Rinaldi S. (): Positive Linear Systems, Theory and Applications. New York: Wiley. Fornasini E. and Valcher M.E. (1997): Recent developments in D positive systems theory. J. App. Math. Comp. Sci., Vol.7, No.4, pp Gantmacher F.R. (1959): The Theory of Matrices. New York: Chelsea. Kaczorek T. (1998a): Weakly positive continuous-time linear systems. Bull. Pol. Acad. Techn. Sci., Vol.46, No., pp Kaczorek T. (1998b): Positive descriptor discrete-time linear systems. Prob. Nonlin. Anal. Eng. Syst., Vol.1, No.7, pp Kaczorek T. (1): Positive 1D and D Systems. London: Springer. Klamka J. (1998): Constrained controllability of positive D systems. Bull. Pol. Acad. Techn. Sci., Vol.46, No.1, pp Ohta Y., Madea H. and Kodama S. (1984): Reachability, observability and realizability of continuous-time positive systems. SIAM J. Contr. Optim., Vol., No., pp Ratajczak Z. (1967): On solutions of the system of homogenous differential linear equations. Scientific Fasicicles of Poznań University of Technology, Mathematics, No.3 pp (in Polish). Rumchev V.G. and James D.J.G. (199): The role of nonnegative matrices in discrete-time mathematical modelling. Int. J. Math. Educ. Sci. Technol., Vol.1, No., pp Rumchev V.G. and James D.J.G. (1995): Spectral characterisation and pole assignment for positive linear discrete-time systems. Int. J. Syst. Sci., Vol.6, No., pp

8 964 T. Kaczorek Valcher M.E. (1996): Controllability and reachability criteria for discrete-time positive systems. Int. J. Contr., Vol.65, No.3, pp Valcher M.E. (1997): On the internal stability and asymptotic behaviour of D positive systems, Part I. IEEE Trans. Circ. Syst., Vol.44, No.7, pp Received: 1 June 1