Int. J. Appl. Math. Comput. Sci. 3 Vol. 3 No. 4 55 59 LOCALLY POSITIVE NONLINEAR SYSTEMS TADEUSZ KACZOREK Institute of Control Industrial Electronics Warsaw University of Technology ul. Koszykowa 75 66 Warszawa Pol e-mail: kaczorek@isep.pw.edu.pl The notion of locally positive nonlinear time-varying linear systems is introduced. Necessary sufficient conditions for the local positiveness of nonlinear time-varying systems are established. The concept of local reachability in the direction of a cone is introduced sufficient conditions for local reachability in the direction of a cone of this class of nonlinear systems are presented. Keywords: local positiveness time-varying nonlinear system local reachability in the direction of a cone. Introduction Roughly speaking positive systems are systems whose trajectories are entirely in the non-negative orthant R n + whenever the initial state input are non-negative. Positive systems arise in the modelling of systems in engineering economics social sciences biology medicine other areas (Alessro Santis 994; Berrman et al. 989; Berrman Plemmons 994; Farina Rinaldi ; Kaczorek ; Rumchev James 99; Rumchev James 999). The single-input single-output externally positive internally positive linear timeinvariant systems were investigated in (Berrman et al. 989; Berrman Plemmons 994; Farina Rinaldi ). The notions of externally positive internally positive systems were extended to singular continuoustime discrete-time two-dimensional linear systems in (Kaczorek ). The reachability controllability of stard singular internally positive linear systems were analysed in (Fanti et al. 99; Klamka 998; Ohta et al. 984; Valcher 996). The notions of weakly positive discrete-time continuous-time linear systems were introduced in (Kaczorek ; ). Recently the positive two-dimensional (D) linear systems were extensively investigated by Fornasini Valcher (Valcher 996; 997) Kaczorek (). Necessary sufficient conditions for the external internal positivities sufficient conditions for the reachability of time-varying linear systems were established in (Kaczorek ; Klamka Kalinowski 998). The notion of the controllability of a dynamic system in the direction of a cone was introduced by Walczak (99) a sufficient condition for the local controllability of nonlinear systems was established. In this paper the notion of local positiveness in the neighborhood of zero of nonlinear time-varying systems will be introduced the necessary sufficient conditions for the local positiveness will be established. The reachability of nonlinear time-varying systems will also be investigated. To the best of the author s knowledge this class of locally positive nonlinear systems has not been considered yet.. Preliminaries Let R n m + be the set of real matrices with non-negative entries R n + : R n +. Consider a nonlinear system described by the equations ẋ f(x u t) x(t ) x (a) y h(x u t) (b) where ẋ dx/dt x R n u R m y R p are the state input output vectors respectively f (x u t) f (x u t) f(x u t). f n (x u t) h(x u t) h (x u t) h (x u t). h p (x u t) ()
56 are R n R p -valued mappings defined on open sets. It is assumed that the functions f (x u t)... f n (x u t) h (x u t)... h p (x u t) are smooth in their arguments i.e. they are real-valued functions of x... x n u... u m t with continuous partial derivatives of any order where x x x... x n T u u u... u m T T denotes transposition. It is also assumed that the system (a) possesses a solution for any admissible input u. where Let A(t) f B(t) f C(t) f D(t) f f( t) h( t) t (3) ẋ A(t)x + B(t)u + N f (x u t) y C(t)x + D(t)u + N h (x u t) x x x x n................ f n f n n u u m................ f n f n u u m h h n................ n h h u u m................ u u m x x x x (4a) (4b) (5) N f (x u t) N h (x u t) are the nonlinear parts of f(x u t) h(x u t) respectively The linear system N f (x u t) lim x u x u N h (x u t) lim x u x u ẋ A(t)x + B(t)u y C(t)x + D(t)u. (6) (7a) (7b) T. Kaczorek is called a linear approximation of the nonlinear system () in the neighborhood of zero (x u ). Example. Consider the nonlinear system Using (5) we obtain A(t) B(t) C(t) D(t) ẋ x t + x + sin x + u + u ẋ x e x + x + u y x + ut + u 3. N f (x u t) f f u f u x h h x h u x x f (x u t) t (8a) (8b) t (9) f (x u t) sin x + u x e x A(t)x B(t)u (a) N h (x u t) h(x u t) C(t)x D(t)uu 3. (b) It is easy to check that the functions in () satisfy the conditions (6). 3. Main Result Lemma. Let ẋ A(t)x () be the linear approximation of the nonlinear autonomous system ẋ f(x t) A(t)x + N f (x t) ()
Locally positive nonlinear systems where A(t) a ij (t) i...n j...n f n f n f n n x x (3) N f (x t) lim. (4) x x If the components f (x t)... f n (x t) of f(x t) satisfy the condition f i (x t) (5) for x j i j x i all t then a ij (t) (6) for i j all t where i j... n. Proof. From (3) we have fi(x t) a ij(t) j x lim x j + f i(... x j... t) f i(... t) x j lim x j + f i(... x j... t) x j (7) since f i (... x j... t). Remark. In the particular case when f(x t) is explicitly independent of time t f(x t) f(x) we have A(t) A the time-invariant matrix (3) is a Metzler matrix satisfying the condition e At R n n + for all t. Definition. The nonlinear system () is called locally positive in the neighborhood of zero (x u ) if there exists a neighbourhood of the zero U such that for any x U R n + we have x(t) U R n + for t (or at least t ε) for some ε > ). Theorem. The nonlinear system () is locally positive in the neighborhood of zero (x u ) if only if f i j (τ) dτ for i j i j... n t f u (t) R n m h x + (t) h u (t) for t. x x R p n + (8a) R p m + (8b) Proof. Note that the condition (8a) is equivalent to 57 a ij (τ) dτ (9) for i j i j... n t. In (Kaczorek ) it was shown that the linear approximation (7) is positive if only if the conditions (9) (8b) are satisfied. Thus based on (6) it is easy to show that the nonlinear system () is locally positive in the neighborhood of zero if only if the linear approximation (7) is positive. Example. (Continuation of Example ) We shall show that the nonlinear system (8) is locally positive in the neighborhood of zero (x u ). The nonlinear system (8) satisfies the conditions (8) since τ dτ dτ dτ dτ for t C(t) R + D(t) t. Therefore by Theorem the nonlinear system (8) is locally positive in the neighborhood of zero. Let C + R n + be a cone in the neighborhood of zero (x u ). Following Walczak (99) the notion of local reachability in the direction of a cone will be introduced. Definition. The nonlinear system () is called locally reachable in the cone direction of a C + if for every state x f C + there exist a time t f t > an input u(t) R m + t t t f such that x(t f ) x f for x(t ) x. A matrix is called the monomial matrix if its every row every column contain only one positive entry the remaining entries are zero. The inverse matrix A of a positive matrix A R n n + is a positive matrix if only if A is a monomial matrix (Kaczorek ).
58 Theorem. The nonlinear system () is locally reachable in the direction of a cone C + R n + if the matrix R f f t Φ(t f τ)b(τ)b T (τ)φ T (t f τ) dτ () t f > t is a monomial matrix. The input that steers the state of the system () in time t f t from x(t ) to the final state x f is given by u(t) B T (t)φ(t f t)r f x f for t t t f. () Proof. If R f is a monomial matrix then there exists the inverse matrix R f R n n + which is also monomial. Hence the output () is well defined u(t) R m + for t t t f. Substituting () into the solution of (7a) for t t f x(t f ) x(t) Φ(t t )x + f t f t t Φ(t τ)b(τ) dτ () x we obtain Φ(t f τ)b(τ)b T (τ)φ T (t f τ)r f x f dτ Φ(t f τ)b(τ)b T (τ)φ T (t f τ) dτ R f x f x f. If the matrix () is a monomial matrix then the linear approximation (7a) of the nonlinear system () is reachable. Thus based on (6) it is easy to show that the nonlinear system () is locally reachable in the cone C +. In particular if the system () is linear from Theorems we obtain the known results given in (Kaczorek ; ; Klamka 998; Klamka Kalinowski 998). 4. Concluding Remarks The notion of local positiveness in the neighborhood of zero of a nonlinear time-varying system was introduced. Necessary sufficient conditions for the local positiveness were established using the linear approximation (7) of the nonlinear system (). The concept of local reachability in the direction of a cone of the positive nonlinear systems () was introduced sufficient conditions for T. Kaczorek local reachability in the direction of a cone using the linear approximation (7) were also derived. With minor modifications the deliberations can be extended to nonlinear discrete time-varying systems. Extensions to singular nonlinear systems D nonlinear systems are open problems. Acknowledgment I wish to express my thanks to Professor Stanisław Walczak for his valuable remarks suggestions. References d Alessro P. de Santis E. (994): Positivness of dynamic systems with non positive coefficients matrices. IEEE Trans. Automat. Contr. Vol. AC-39 pp. 3 34. Berrman A. Neumann M. Stern R. (989): Nonnegative Matrices in Dynamic Systems. New York: Wiley. Berman A. Plemmons R.J. (994): Nonnegative Matrices in Mathematical Sciences. Philadelphia: SIAM Press. Caccetta L. Rumchev V.G. (998): Reachable discrete-time positive systems with minimal dimension control sets. Dynam. Cont. Discr. Impuls. Syst. Vol. 4 pp. 539 55. Caccetta L. Rumchev V.G. (): A survey of reachability controllability for positive linear systems. Ann. Opers. Res. Vol. 98 pp.. Fanti M.P. Maione B. Turchsano B. (99): Controllability of multi-input positive discrete-time systems. Int. J. Contr. Vol. 5 No. 6 pp. 95 38. Farina L. Rinaldi S. (): Positive Linear Systems Theory Applications. New York: Wiley. Fornasini E. Valcher M.E. (997): Recent developments in D positive systems theory. J. Appl. Math. Comp. Sci. Vol. 7 No. 4 pp. 73 735. Gantmacher F.R. (959): The Theory of Matrices. Chelsea New York. Kaczorek T. (): Externally internally positive timevarying linear systems. Int. J. Appl. Math. Comput. Sci. Vol. No. 4 pp. 957 964. Kaczorek T. (): Positive D D Systems. London: Springer-Verlag. Kaczorek T. (998): Weakly positive continuous-time linear systems. Bull. Pol. Acad. Sci. Techn. Sci. Vol. 46 No. pp. 33 45. Kaczorek T. (998): Positive descriptor discrete-time linear systems. Probl. Nonlin. Anal. Eng. Syst. Vol. No. 7 pp. 38 54.
Locally positive nonlinear systems Klamka J. (998): Constrained controllability of positive D systems. Bull. Pol. Acad. Techn. Sci. Vol. 46 No. pp. 93. Klamka J. Kalinowski J. (998): Reachability controllability of positive continuous-time linear systems with time variable coefficients. Bull. Pol. Acad. Techn. Sci. Vol. 49 No. 4 pp. 633 64. Ohta Y. Madea H. Kodama S. (984): Reachability observability realizability of continuous-time positive systems. SIAM J. Contr. Optim. Vol. No. pp. 7 8. Rumchev V.G. (): Feedback positive feedback holdability of discrete-time systems. Syst. Sci. Vol. 6 No. 3 pp. 5 3. Rumchev V.G. Anstee R.P. (999): Asymptotic reachable sets for discrete-time positive linear systems. Syst. Sci. Vol. 5 No. pp. 4 47. 59 Rumchev V.G. James D.J.G. (99): The role of nonnegative matrices in discrete-time mathematical modelling. Int. J. Math. Educ. Sci. Technol. Vol. No. pp. 69 8. Walczak S. (984): A note on the controllability of nonlinear systems. Math. Syst. Theory Vol. 7 pp. 35 356. Valcher M.E. (996): Controllability reachability criteria for discrete-time positive systems. Int. J. Contr. Vol. 65 No. 3 pp. 5 536. Valcher M.E. (997): On the internal stability asymptotic behaviour of D positive systems. IEEE Trans. Circ. Syst. Part I Vol. 44 pp. 6 63. Received: 5 May 3 Revised: 8 July 3 Rumchev V.G. James D.J.G. (995): Spectral characterisation pole assignment for positive linear discrete-time systems. Int. J. Syst. Sci. Vol. 6 No. pp. 95 3.