DISSIPATION CONTROL OF AN N-SPECIES FOOD CHAIN SYSTEM

Transcription

1 INTERNATIONAL JOURNAL OF INFORMATION AND SYSTEMS SCIENCES Volume 1 Number 3-4 Pages c 005 Institute for Scientific Computing and Information DISSIPATION CONTROL OF AN N-SPECIES FOOD CHAIN SYSTEM LICHUN ZHAO 1 QINGLING ZHANG AND QICHANG YANG 1 Abstract First the passive control and H control of a n-species food chain system are studied and some sufficient conditions rendering the closed-loop system globally asymptotically stable at the positive equilibrium point are given Second an optimal control law of a n-species food chain system is acquired Key Words Passivity analysis Dissipation-control Optimal control H - control L - gain Food chain system 1 Introduction The management of population resources is one of the important control problems for ecosystems Much attention has been paid to the optimal control of population systems [1 6] The occurrence of impulse phenomenon arouses some researchers interests on the impulse control of population sys-tems [7 10] Recently induction control was also applied to the control of population systems [11] In general there are different controls exerted on the population systems to investigate their permanence and stability etc Since most biological systems in the natural world have nonlinearity it is this nonlinearity that makes the system in question have dissipative structure thus Prigogine s dissipative structure theory plays a key role in dealing with these problems in nonlinear biological systems As two special cases of dissipation passivity analysis and H - control are investigated by many researchers [1 17] In this paper we will discuss their applications in population dynamic systems In the natural world the relations of populations in ecosystems are very complicated In these relations the food chain system is typical Now consider the following n-species food chain system (11) ẋ 1 = x 1 (a 10 a 11 x 1 a 1 x ) x 1 (0) = x 10 > 0 ẋ = x ( a 0 + a 1 x 1 a 3 x 3 ) x (0) = x 0 > 0 ẋ n 1 = x n 1 ( a n 10 + a n 1n x n a n 1n x n ) x n 1 (0) = x (n 1)0 > 0 ẋ n = x n ( a n0 + a nn 1 x n 1 ) x n (0) = x n0 > 0 where a 11 is a non-negative constant a ij (i = 1 n; j = 1 n) is positive constant x i (t)(i = 1 n) is the population of the i-th species In ecology the first species is said to be the first producer the second species the first consumer Received by the editors June and in revised form October 004 This work is Supported by Natural Science Foundation of People s Republic of China ( ) ( ) and ( ) 48

2 SEVERAL CONTROLS OF AN N-SPECIES FOOD CHAIN SYSTEMS 49 and the third species the secondary consumer and so on The study for n-species the food chain systems is complete Some results of their stability and permanence are seen in [1 ] The food chain systems with controls are studied in some papers such as [16] This paper is organized as follows: In section we study the global stability of a n-species chain food system and get stability conditions for it In section 3 we investigate the H - control problem of system (11) some sufficient conditions which make the system in question globally stable and permanent are obtained In section 4 Using passivity control we study the optimal control problem and get optimal control law which makes the system permanent Passivity based control 1 Preliminaries Consider a nonlinear system of the form { ẋ = f(x) + g(x)u (1) y = h(x) with state space X = R n set of inputs U = R m and set of outputs Y = R p For this system the following concepts are given Definition 1 [15] System (1) is said to be dissipative with supply rate s(u(t) y(t)) if there exists a storage function S : X R + such that for all x(t 0 ) = x 0 X t t 0 and all the input function u the following inequality is satisfied () S(x(t)) S(x(t 0 )) + t t 0 s(u(t)y(t))dt Inequality () is called dissipation inequality (where x(t t 0 x 0 ) is the solution of system (1) with initial condition x 0 X) Definition [15] System (1) is said to be passive if it is dissipative with supply rate u T y System (1) is said to be strictly input passive if there exists a σ > 0 such that the system is dissipative with supply rate s(uy) = u T y σ u System (1) is said to be strictly output passive if there exists a ǫ > 0 such that the system is dissipative with supply rate s(uy) = u T y ǫ y Definition 3 [15] If u 0 y 0 t 0 imply lim t x(t) = 0 t 0 then system (1) is said to be zero state detectable Differentiate the inequality () with respect to t at t 0 we have (3) S x (x)f(x) s(uh(x)) x u where S x (x) = ( S x 1 S x n ) = (S 1 S n ) Inequality (3) is called differential dissipation inequality

3 430 L ZHAO Q L ZHANG AND Q YANG Lemma 1 [15] In system (1) for ǫ 1 > 0 (4) S x (x)[f(x) + g(x)u] u T h(x) ǫ 1 h T (x)h(x) x u is equivalent to (5) S x (x)f(x) ǫ 1 h T (x)h(x) S x (x)g(x) = h T (x) The following inequality is often used (6) S x (x)[f(x) + g(x)u] 1 γ u 1 h(x) x u where γ is a positive constant Lemma [15] For system (7) ẋ = f(x) + g(x)k(x) f(0) = 0 k(0) = 0 assume that the equilibrium x = 0 of ẋ = f(x) is asymptotically stable and there exists a C 1 function S(x) > 0 for some ǫ > 0 S(x) is semi positive at point x = 0 and satisfies (8) S x (x)[f(x) + g(x)k(x)] ǫ k(x) then the equilibrium x = 0 of system (7) is asymptotically stable For system (11) the existence and uniqueness conditions of the positive equilibrium of system (11) is given in the following lemma Lemma 3 [1] If system (11) has a unique isolated positive equilibrium x = (x 1 x x 3) then it is globally asymptotically stable Lemma 4 For system (11) if the conditions in Lemma 3 hold the equilibrium x = (x 1 x x 3) of system (11) is globally asymptotically stable Passivity based-control System (11) is equivalent to the following system (x is a positive equilibrium point of system (11)) ẋ 1 = x 1 [ a 11 (x 1 x 1) a 1 (x x )] x 1 (0) = x 10 > 0 ẋ = x [a 1 (x 1 x 1) a 3 (x 3 x 3)] x (0) = x 0 > 0 (9) ẋ n = x n a nn 1 (x n 1 x n 1) x n (0) = x n0 > 0 Because the producer of a food chain system plays important role in the permanence problem of the system we track its density The matrix form of corresponding control system with output y is (10) { ẋ = f (x) + g (x)u (x) y = h (x)

4 SEVERAL CONTROLS OF AN N-SPECIES FOOD CHAIN SYSTEMS 431 where f (x) = x 1 [ a 11 (x 1 x 1) a 1 (x x )] x [a 1 (x 1 x 1) a 3 (x 3 x 3)] x n [a nn 1 (x n 1 x n 1 )] u (x) = u 1 (x) u (x) u n (x) g (x) = diag{g i (x)}(i = 1 n) Theorem 1 For system (10) if the conditions in Lemma 3 hold and g 1 (x) = x 1 g (x) = g 3 (x) = = g n (x) 0 then system (10) is strictly output passive Proof In system (10) consider a storage function n (11) S(x) = α i (x i x i lnx i ) where i=1 α 1 = 1 α = a 1 a 1 α 3 = a 1a 3 a 1 a 3 α n = a 1a 3 a n 1n a 1 a 3 n nn 1 We show that system(10) is dissipative with supply rate s(u y ) = u T y ǫ y (ǫ > 0) Obviously S(x) satisfies the second inequality of system (5) And for 0 < ǫ a 11 we can get (1) a 11 (x 1 x 1) ǫ(x 1 x 1) So the first equation of (5) holds provided that 0 < ǫ a 11 According to Lemma 1 and Definition we complete the proof of the theorem Now we search the control law which asymptotically stabilizes the equilibrium x = x We can get the following system by substitution N = x x (13) { Ṅ = ˆf (N) + ĝ (N)û (N) y = ĥ(n) where ˆf (N) = f (N + x ) ĝ (N) = g (N + x ) û (N) = u (N + x ) ĥ (N) = h (N + x ) For system (13) we can get the following theorem Theorem For system (13) suppose (1) The conditions in Theorem 1 hold; () there exists a ε > 0 and ε a 11 ; then the control law û (N) = û 3 (N) = = û n (N) 0 û (1) 1 = N 1 û 1 û () 1 ǫ = N 1 + ǫ (where = N1 + 4ǫa 11 N1) asymptotically stabilizes the equilibrium N = 0 of system (13) Further if û (N) 0 then the equilibrium N = 0 of system

5 43 L ZHAO Q L ZHANG AND Q YANG (13) is globally asymptotically stable Proof Take a storage function as follows n n (14) Ŝ(N) = [α i (N i + x i x i lnn i + x i )] ln α i ( e x ) x i i=1 i=1 i α i (i = 1 n) is same as those aforementioned Obviously Ŝ(N) > 0 Ŝ(0) = 0 Denote k = û (N) = û S x = ŜN(N) = (Ŝ1 Ŝ Ŝ 3 ) f = ˆf g = ĝ in (8) we get a 11 N1 + N 1 û 1 ǫû 1 0 ǫû (15) 0 ǫû n According to given conditions û (N) = = û n (N) 0 we can see that all inequalities are satisfied except the first one in (15) Change the first one into (16) ǫû 1 + N 1 û 1 a 11 N 1 0 If û 1 satisfies û (1) 1 = N 1 û 1 û () 1 ǫ = N 1 + ǫ (where = N 1 +4ǫa 11 N 1) then inequality (16) holds According to Lemma 3 the equilibrium N = 0 of system (13) is asymptotically stable On the other hand because û satisfies Ŝ N (N)[ˆf (N) + ĝ (N)û ] ǫ û If u (N) 0 and Ŝ(N) is proper function Ŝ(N) is a Lyapunov function Further we can obtain that the equilibrium N = 0 of closed-loop system is globally asymptotically stable Thus we complete the proof of the theorem The global stability of the equilibrium point N = (0 0) of system (13) is equivalent to the global stability of the equilibrium point x = (x 1 x x n) of system (9) thus we know that the controls in Theorem make the system in question permanent However if systems are disturbed by outside signals unknown but bounded in L space we may study it by H -control This is our next topic 3 H -control 31 Preliminaries Consider the following nonlinear system with standard control structure (Figure 1) ẋ = f(xud) (31) : y = g(xud) z = h(xud) where u is the control input d is the disturbance input y is the measured output z is the controlled output

6 SEVERAL CONTROLS OF AN N-SPECIES FOOD CHAIN SYSTEMS 433 Figure 1 The closed-loop system with standard control structure Optimal H -control problem is: to design a control law based on y such that the L -gain from the disturbance input d to the controlled output z of the closed-loop system is minimum and the closed - loop system is stable in some sense In fact it is very difficult to realize the optimal H control So we look for suboptimal H -control ie for a prescribed value γ(> 0) to find a controller such that the L - gain of the closed-loop system is not larger than γ(> 0) And then γ(> 0) is decreased continuously to achieve the optimal H control Suppose that there is an external disturbance input and all state - vector can be measured ie y = x in system (1) thus a simpler form of system (31) is (3) ẋ = ( f(x) + b(x)u ) + g(x)d x z = u where all functions belong to C k (k ) According to Theorem 711 in [15] for system () the control u such that the L -gain from the disturbance input d to the controlled output z is not larger than γ(> 0) was found by solving P 0 in the following Hamilton - Jacobi inequality (HJIa) : P x f(x) + 1 P x[ 1 γ g(x)gt (x) b(x)b T (x)]p T x + 1 xt x 0 where P x = (P 1 P P 3 ) 3 H -control The matrix form of the control system of system (11) with disturbance is described as ẋ = f 3 (x) + b 3 (x)u 3 (x) + g 3 (x)d 3 (x) u 3 R m d 3 R r (33) ( ) h3 (x) y 3 = x X y u 3 (x) 3 R s where x 1 (a 10 a 11 x 1 a 1 x ) x ( a 0 + a 1 x 1 a 3 x 3 ) f 3 (x) = x n 1 ( a n 10 + a n 1n x n a n 1n x n ) x n ( a n0 + a nn 1 x n 1 )

7 434 L ZHAO Q L ZHANG AND Q YANG b 3 (x) = diag{b 3i (x)} h 3 (x) = (x 1 x 1 x n x n) T d 3 (x) = {d 31 (x) d 3n (x)) T where i = 1 n If system (11) has a unique positive equilibrium point x By substitution N = x i x i (i = 1 n) system (33) is changed into the following form (34) Ṅ = f 3 (N + x ) + b 3 (N + x )u 3 (N + x ) + g 3 (N + x )d 3 (N + x ) y 3 = ( h3 (N + x ) u 3 (N + x ) where f 3 (N + x ) = ) (a 11 N 1 + a 1 N )(N 1 + x 1) (a 1 N 1 a 3 N 3 )(N + x ) (a n 1n N n a n 1n N n )(N n 1 + x n 1) a nn 1 N n 1 (N n + x n) Other matrices have the same substitution and the following theorem is got easily Theorem 31 The global stability of the equilibrium point N = (0 0 0) of system (33) is equivalent to the global stability of the equilibrium point x = (x 1 x x n) of system (34) Using Theorem 711 and Proposition 71 in [15] we look for the control law which guarantee the stability of original system Theorem 3 In system (34) suppose that system (11) has a unique positive equilibrium x and for γ > 1 (i) If 0 < a 11 < 1 take b 31(N + x ) = 1 + (1 a 11 )(N 1 + x 1 ) b 3i (N + x ) = α i + (N i + x i ) /α i where α i (i = 1 n) is the same as aforementioned let a control law be u 3 (N + x ) = b T 3 (N + x )P T N (N + x ) = γn 1 N 1 + x 1 + (1 a 11 )(N 1 + x 1 ) 1 γn N + x α + (N + x ) γn n α N n + x n + (N n + x n) n

8 SEVERAL CONTROLS OF AN N-SPECIES FOOD CHAIN SYSTEMS 435 (ii) If a 11 1 take b 31(N + x ) = 1 b 3i (N + x ) = α i + (N + x ) /α i (i = 1 n) let a control law be u 3 (N + x ) = b T 3 (N + x )P T N (N + x ) = γn 1 N 1 + x 1 γn N + x α + (N + x ) γn n α N n + x n + (N n + x n) n then the L - gain of the closed - loop of system (39) from d to z is not larger than γ Proof Let (35) P(N + x ) = γ[ Thus P(N + x ) N i n α i (N i + x i x i ln(n i + x i ))] i=1 = γn iα i N i + x i = 1 n i P(N + x ) ) T = (P 1 P P n ) T Denote P N (N + x ) = ( P(N + x ) N 1 N n and substitute f 3 (N + x ) P N (N + x ) b 3 (N + x ) g 3 (N + x ) h 3 (N + x ) into (HJIa) we get (36) a 11 N 1 (N 1 + x 1)P 1 a 1 N (N 1 + x 1)P 1 + ( 1 γ b 31(N + x ))P1 + N1 +a 1 N 1 (N + x )P a 3 N 3 (N + x )P + ( 1 γ b 3(N + x ))P + N +a 3 N (N 3 + x 3)P 3 a 34 N 4 (N 3 + x 3)P 3 + ( 1 γ b 33(N + x ))P3 + N3 +a n 1n N n (N n 1 + x n 1)P n 1 a n 1n N n (N n 1 + x n 1)P n 1 +( 1 γ b 3n 1(N + x ))P n 1 + N n 1 +a nn 1 N n 1 (N n + x n)p n + ( 1 γ b 3n(N + x ))P n + N n 0 Then we have b 31(N + x γn 1 )( N 1 + x ) + N1 + ( 1 N 1 + x ) a 11 N1γ 1 N 1 (37) b 3(N + x )( γn α N 1 + x ) + N + ( N α N + x ) b 3n(N + x )( γn nα n N n + x ) + Nn + ( N nα n n N n + x ) 0 n

9 436 L ZHAO Q L ZHANG AND Q YANG If the following inequalities hold b 31(N + x )γ N 1 ( N 1 + x ) + N N ( 1 N 1 + x ) a 11 N1γ 0 1 (38) b 3(N + x )γ ( N α N 1 + x ) + N + ( N α N + x ) 0 b 3n(N + x )γ ( N nα n N n + x ) + Nn + ( N nα n n N n + x ) 0 n then the inequalities in (37) hold According to the inequalities in (38) we have b 31(N + x )γ + a 11 (N 1 + x 1) γ [1 + (N 1 + x 1) ] 0 (39) b 3(N + x )γ α [α + (N + x ) ] 0 b 3n(N + x )γ αn [αn + (N n + x n) ] 0 If γ > 1 and b 31 (N + x ) b 3 (N + x ) b 3n (N + x ) satisfy (310) γ > 1 a 11(N 1 + x 1) + [a 11 (N 1 + x 1 ) ] + [1 + (N 1 + x 1 ) ]b 31 (N + x ) b 31 (N + x ) γ > 1 α + (N + x ) b 3 (N + x )α Let (311) γ > 1 b 31 (N + x ) = α n + (N n + x n) b 3n (N + x )α n = 1 + (1 a 11 )(N 1 + x 1 ) a 11 < 1 = 1 a 11 1 b 3 (N + x ) = α + (N + x ) /α b 3n (N + x ) = α n + (N n + x n) /α n According to theorem 711 in [15] we reach the conclusion Thus complete the proof of the theorem Theorem 33 System (4) without output disturbance is Ṅ = f 3 (N + x ) + b 3 (N + x )u 3 (N + x ) u 3 R m d 3 R r (31) ( h3 (N + x ) ) y 3 = u 3 (N + x x X y ) 3 R s

10 SEVERAL CONTROLS OF AN N-SPECIES FOOD CHAIN SYSTEMS 437 where f 3 (N + x ) b 3 (N + x ) u 3 (N + x ) h 3 (N + x ) are expressed as the previous If x is a unique equilibrium of system (11) for γ > 0 the control law (313) u 3 (N + x ) = b T 3 (N + x )P T N(N + x ) globally asymptotically stabilizes the equilibrium N = 0 of system (314) Ṅ = f 3 (N + x ) b 3 (N + x )b T 3 (N + x )P T N(N + x ) The proof of the theorem is the same as that of proposition 71 in [15] so we omit it Further we can get the following corollary Corollary 31 If the conditions in Theorem 43 hold then the control law (313) makes the original system (33) corresponding to the closed - loop system (314) permanent 4 Optimal control Consider the following system (41) ẋ = f(x) + g (x)u f(0) = 0 x(0) = x 0 Lemma 41 [15] For system (41) given the following performance (4) min 0 ( u(t) + l(x(t)))dt where l 0 is a cost function and l(0) = 0 If system ẋ = f(x) y = l(x) is zero - state detectable and there exists a nonnegative solution V (x) to the following equation (43) V x (x)f(x) 1 V x(x)g (x)g T (x)vx T (x) + 1 l(x) = 0 V (0) = 0 then control u = α(x) = g T (x)v T x (x) is the optimal control law of system (41) with performance (4) By substitution N = x x system (11) is changed into (44) Ṅ = f 4 (N) The corresponding control system is (45) Ṅ = f 4 (N) + g 4 (N)u 4 (N) where f 4 (N) = (a 11 N 1 + a 1 N )(N 1 + x 1) (a 1 N 1 a 3 N 3 )(N + x ) (a n 1n N n a n 1n N n )(N n 1 + x n 1) a nn 1 N n 1 (N n + x n) g 4 (N) = diag{g 41 (N)} u 4 (N) = (u 41 (N) u 4n (N)) T Using the result of Lemma 41 we discuss the optimal control problem of system (45) with the following performance: (46) min 0 [ u 4 (N) + n p i Ni (t)]dt i=1

11 438 L ZHAO Q L ZHANG AND Q YANG where p i > 0(i = 1 n) is the unit cost of the i-th population in system (45) respectively Thus the second term of performance is the total cost The first term is the cost of the control Next we will describe the optimal problem with system (45) and performance (46) First we look for the non-negative solution of equation (44) Let V x (x) = (V 1 V V 3 ) (where V = n i=1 α i[n i + x i x i ln(n i + x i ) ln( e x ) x i ]) and put it i into the equation (43) we get a 11 N 1 α 1N 1 (N 1 + x 1 ) g 41(N) p 1 N 1 (47) (48) α N (N + x ) g 4(N) p N α nn n (N n + x n) g 4n(N) p n Nn = 0 a 11 N 1 α 1N 1 (N 1 + x 1 ) g 1(N) p 1 N 1 = 0 α N (N + x ) g (N) p N = 0 Let (49) g 41 = α nn n (N n + x n) g n(n) p n Nn = 0 p1 a 11 α 1 (N 1 + x 1) g 4 = p α (N + x ) g 4n = p n (N n + x n ) α Thus we can obtain the following control law u 41 = p 1 a 11 N 1 u 4 = p N (410) u 4n = p n N n According to Theorem 41 we know that the control law (45) is the optimal control law of system (45) with performance (46) and it makes system (45) asymptotically stable From the discussion above the following theorem are obtained Theorem 41 For system (45) with performance (46) the control law (410) is the optimal control law and asymptotically stabilizes the equilibrium N = 0 of it

12 SEVERAL CONTROLS OF AN N-SPECIES FOOD CHAIN SYSTEMS 439 In short the optimal control problem investigated by passivity analysis is more convenient than that investigated by maximum principle 5 Conclusions Because most biological systems have dissipative structure dissipative analysis plays an important role in biological control theory In this paper we use dissipative analysis to investigate passivity control H control and optimal control and get some control laws which make systems in question stable These give control theory more access to biology systems and at the same time biological systems provide wider space for the applications of control theories References [1] Lansun Chen Models and research of mathematical ecology Science Press Beijing 1988 [] Bean-San Goh Management and analysis of biological populations Elsevier Scientific Publishing Company New York 1980 [3] Colin W ClarkThe Optimal management of renewable resources (Second Edition) John Wiley and Sons Inc New York 1990 [4] Xinyu Song Lansun Chen Optimal harvesting and stability of a twospecies competitive system with stage structure Mathematical Biosciences 17(001) [5] Mang Fan Ke Wang Optimal harvesting policy for a single population with periodic coefficients Mathematical Biosciences 15(1998) [6] Svetoslav I Nenov Impulsive controllability and optimization problems in population dynamics Nonlinear Analysis 36 (1999) [7] J Angelova A Dishliev Optimization problems for one-impulsive models from population dynamics Nonlinear Analysis 39 (000) [8] Xinzhi Liu Katrin Rohlf Impulsive control of a Lotka-Volterra system IMA Journal of Mathematical Control and Information 15 (1998) [9] Xinzhi Liu Impulsive stabilization and application to population growth models Rocky Mountain Journal of Mathematics 5(1) (1995) [10] Yosef Cohen Applications of optimal impulse control to optimal foraging problems Lecture Notes in Biomathematics 73 Springer Verlag Berlin 1987 [11] S V Emel yanov I A Burovoi and F Yulevada Control of indefinite nonlinear dynamic systems (Induced internal feedback) Lecture Notes in Control and Information Science 31 Springer-Verlag London 1998 [1] Xinping Guan Changchun Hua and Yinggan Tang Robust passivity control for a class of nonlinear systems Control and Decision 16(5) (001) [13] Chunbo Feng Passivity analysis of feedback systems and its application Acta Automatica Sinica11() (1985) [14] Chunbo Feng Stability analysis for time - varying nonlinear systems via passivity analysis Acta Automatica Sinica (3) (1997) [15] Arjan van der schaft L -Gain and passivity techniques in nonlinear control Springer-Verlag London 00 [16] Haiying Jing The stability and permanence n-species food chain systems with feedback controls Journal of Biomathematics 17(1) (00) 1-0

13 440 L ZHAO Q L ZHANG AND Q YANG [17] Yang Xia Jufang Chen Permanence and existence of positive periodic solution for diffusive Lotka-Volterra model Journal of Biomathematics 1(1) (1997) Anshan Normal University Anshan P R China College of Sciences Northeastern University Shenyang PR China