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On the stability of homogeneous nuclear reactors
Journal of The Franklin Institute DEVOTED T O S C I E N C E A N D T H E M E C H A N I C A R T S
Volume
284, Number
5
November
1967
On the Stability of Homogeneous Nuclear Reactors b y s. ~Bm~TONI, A. C E L L ~ A a n d G. P. SZEGO Istituto di Fisica Universitdt di Milano, Milano, Italia ABSTRACT: In this paper a homogeneous nuclear reactor is considered, with the usual substitutions, as an indirect control system of the Lur' e type. A new Liapunov function is presented that takes into account a property of the derivative of the nonlinearity. From this an improved stability criterion is derived. Introduction Consider a nuclear reactor without delayed neutrons described by the followhag set of equations du/dt =Gu d- av d~/dt = (k/e) v (1) k=ko-r'u-~ where u is a real n vector, whose components are the temperatures of the different constituents of the reactor G is a non-singular n X n matrix is a positive scalar, the prompt neutron's density k is a scalar, the reactivity a, r are n-vectors is a scalar, representing the coupling with an external control system. For this model see, for instance, (1, 2, 3). If ]co ~ 0, system (1), as is well known, has two equilibrium points; i.e., ~/41 ~
0
(2) 1)1 ~ 0
279
S. Albertoni, A. CeUina, G. P. Szego
and u2 = - k o G - t a / (~ - r ' G - l a ) v2 = ko/ (~ -- r ' G - l a ) .
(3)
We take advantage of the usual transformation = In v/v,
(4) (5)
in order to have the system in the form of a general indirect control system ?c = Gx q- av ir = -- ( 1 / l ) r ' x -
(6)
(~/1)v
v = ¢(a) = v2(e" -
1).
I t is important to note that the change in variables (4) and (5) has the effect of lifting the equilibrium point (2) to infinity. We now construct a new Liapunov function for system (6) and from it derive a new stability criterion. The following lemmas are therefore necessary: Lemma 1. E x t e n s i o n Theorem (4) If v = x ( x ) , w = ¢(x) are real functions of real variables in R" and:
i) ii) iii) iv) v)
v = x(x) CC 2 ¢(x) = 0 forx = 0 &(x) ~ 0 , x ~ R ~, x ~ O There exists no {xn}, x, ER', ~b(x) = (grad x ( x ) ' f ( x ) )
1[ xn ]l--~ ~ such that ~k(x~) --*0
Then if the equilibrium point x = 0 of ~ = f ( x ) is locally asymptotically stable, it is also globally asymptotically stable. 2. K a l m a n - I a c u b o v i c h L e m m a (1, 7) Given the stable matrix G, a symmetric matrix D > 0, vectors a and ( r / 2 l ) - $ a ( r / 2 l ) and scalars ~ , ~ > 0, then a necessary and sufficient condition for the existence of a solution such as a matrix H (necessarily > 0) and a vector q of the system, Lemma
H G - 4 - G ' H = - qq' - eD Ha( r / 2 l ) 4 - ~ G ( r / 2 l ) = "yq -- ( ~ / l ) r ' a = 3,2 > 0
(7)
is that e be small enough and that the relation -
(5/1)r'a -f- 2 Re [ ( r ' / 2 l ) ( [ -
~G) ( i o J -
G)-la3 > 0
(8)
be satisfied for all real o:. T h e o r e m I . Assume that: i) The matrix of system (6) is stable; ii) The system is asymptotically stable in linear approximation; iii) There exists a
280
Journal of The Franklin Institute
On the Stability of Homogeneous Nuclear Reactors real number 6 ~ 0, such that Re (1 - i ~ ) F ( i ~ ) < 0
(9)
where
F(i~) = -- ~ - (r'/1) (loci - G)-la
(10)
then the equilibrium point x = 0 of system (6) is globally asymptotically stable.
Proof: Consider the function (a generalized quadratic form of x and ~(~)) ~, = x'tIx + f f O(~-)d~-+ 6~ ~r r. x + ~6 ~~" ¢2;
6 > 0,
(6)
thus
= x'(HG ~ - G ' H ) x +2c~x'
Ha-~+6G 2
-
dp2( ~
r'ia )
- 6 - -
O¢(r';x
-~o-~
- - +
~.¢)
"
(7)
We have 0¢/0a > 0, thus for Lemma 2 ~ will be negative definite if the condition,
-- (6/1)r'a ~- 2 Re [-r'/2l) (I -- 6G) (i¢oi - a)-'a-] > 0
(8)
is satisfied. Taking advantage of the identity: -- 6(iwI - G)-~(io~I - G) + 61 = O. It is possible after a few computations to rewrite this last condition in the form Re (1 - 6i~)F(io~) < 0
(9)
for all ~, where F(i~) is given by Eq. 10. Under such conditions, the function is then negative definite with respect to (x = 0, z = 0). The application of Lemma 1) then completes the proof. Recalling the transformation (4), this theorem then assures that the equilibrium point (2) is asymptotically stable for system (1), with the region of attraction being the region ~ > O. Of course it is not true that every solution of system (1) other than (2) approaches (3) as t --~ ~ : in fact, it is possible to prove (6) that there is at least one trajectory joining (2) with ~. The preceding theorem then assures that such a trajectory lies in the v _< 0 region and thus has no physical meaning. For the given system, the original Popov criterion is, (1, 5, 8)
(E/1) (~ - r'G-la) + (l/l) Re (1 + fli~)[r'G-i(i~I - G)-la~ > O.
(10)
It is an open question to investigate the relationships between these two conditions and to decide whi(~h particular problem gives the larger region of stability.
Vol. 284, No. 5, November 1967
281
,S. Albertoni, A. Cellina, G. P. Sze~o
~i'@/ICP.8 (I) 8. I.efl~etm, "Stability of Nonlinear Control Systems," New York, Academic Press, 1965. (2) W. IMran and K. Meyer, "Effect of Delayed Neutrons on the Stability of a Nuclear Power Reactor," N.S.E., Vol. 24, No. 4, p. 356, 1966. (3) S. Albertoni, A. Cellina and G. P. S~go, "Criteri di stabilitS, di reattori nucleari," Milano, Italy, Tamburini Editore, 1966. (4) G. P. Szego, "New Tlieorema on Stability and Attraction and their Application to a Control Problem," From Stability Problems of Solutions of Differential Equations, Editori "Oderisi" Gubbio (Italy), pp. 107-175, 1966. (5) Y. H. Ku and H. T. Chieh, "Extension of Popov's Theorcm for Stability of Nonlinear Control Systems," Jour. Frank. Inst., Vol. 279, No. 6, pp. 401-416, 1965. (6) N. P. Bhatia and G. P. Szego, "Dynamical System; Stability Theory and Applications," Lecture Notes on Mathematics, Springer-Verlag, (In print). (7) M. A. Aizerman and F. R. Gantmacher, "Absolute Stability of Regulator Systems," Sail Franeisto, Calif., Holden-Day, Inc. 1.964. (8) Y. H. Ku and H. T. Chieh, "New Theories oil Absolute Stability of Non-Autonomous Noalincar Control Systems," IEEE Itder~at. (YoJw. Rcc., Vol. 14, Pt. 7, pp. 260-271, 1966.
282
Journal of The Franklin Institute
Volume
284, Number
5
November
1967
On the Stability of Homogeneous Nuclear Reactors b y s. ~Bm~TONI, A. C E L L ~ A a n d G. P. SZEGO Istituto di Fisica Universitdt di Milano, Milano, Italia ABSTRACT: In this paper a homogeneous nuclear reactor is considered, with the usual substitutions, as an indirect control system of the Lur' e type. A new Liapunov function is presented that takes into account a property of the derivative of the nonlinearity. From this an improved stability criterion is derived. Introduction Consider a nuclear reactor without delayed neutrons described by the followhag set of equations du/dt =Gu d- av d~/dt = (k/e) v (1) k=ko-r'u-~ where u is a real n vector, whose components are the temperatures of the different constituents of the reactor G is a non-singular n X n matrix is a positive scalar, the prompt neutron's density k is a scalar, the reactivity a, r are n-vectors is a scalar, representing the coupling with an external control system. For this model see, for instance, (1, 2, 3). If ]co ~ 0, system (1), as is well known, has two equilibrium points; i.e., ~/41 ~
0
(2) 1)1 ~ 0
279
S. Albertoni, A. CeUina, G. P. Szego
and u2 = - k o G - t a / (~ - r ' G - l a ) v2 = ko/ (~ -- r ' G - l a ) .
(3)
We take advantage of the usual transformation = In v/v,
(4) (5)
in order to have the system in the form of a general indirect control system ?c = Gx q- av ir = -- ( 1 / l ) r ' x -
(6)
(~/1)v
v = ¢(a) = v2(e" -
1).
I t is important to note that the change in variables (4) and (5) has the effect of lifting the equilibrium point (2) to infinity. We now construct a new Liapunov function for system (6) and from it derive a new stability criterion. The following lemmas are therefore necessary: Lemma 1. E x t e n s i o n Theorem (4) If v = x ( x ) , w = ¢(x) are real functions of real variables in R" and:
i) ii) iii) iv) v)
v = x(x) CC 2 ¢(x) = 0 forx = 0 &(x) ~ 0 , x ~ R ~, x ~ O There exists no {xn}, x, ER', ~b(x) = (grad x ( x ) ' f ( x ) )
1[ xn ]l--~ ~ such that ~k(x~) --*0
Then if the equilibrium point x = 0 of ~ = f ( x ) is locally asymptotically stable, it is also globally asymptotically stable. 2. K a l m a n - I a c u b o v i c h L e m m a (1, 7) Given the stable matrix G, a symmetric matrix D > 0, vectors a and ( r / 2 l ) - $ a ( r / 2 l ) and scalars ~ , ~ > 0, then a necessary and sufficient condition for the existence of a solution such as a matrix H (necessarily > 0) and a vector q of the system, Lemma
H G - 4 - G ' H = - qq' - eD Ha( r / 2 l ) 4 - ~ G ( r / 2 l ) = "yq -- ( ~ / l ) r ' a = 3,2 > 0
(7)
is that e be small enough and that the relation -
(5/1)r'a -f- 2 Re [ ( r ' / 2 l ) ( [ -
~G) ( i o J -
G)-la3 > 0
(8)
be satisfied for all real o:. T h e o r e m I . Assume that: i) The matrix of system (6) is stable; ii) The system is asymptotically stable in linear approximation; iii) There exists a
280
Journal of The Franklin Institute
On the Stability of Homogeneous Nuclear Reactors real number 6 ~ 0, such that Re (1 - i ~ ) F ( i ~ ) < 0
(9)
where
F(i~) = -- ~ - (r'/1) (loci - G)-la
(10)
then the equilibrium point x = 0 of system (6) is globally asymptotically stable.
Proof: Consider the function (a generalized quadratic form of x and ~(~)) ~, = x'tIx + f f O(~-)d~-+ 6~ ~r r. x + ~6 ~~" ¢2;
6 > 0,
(6)
thus
= x'(HG ~ - G ' H ) x +2c~x'
Ha-~+6G 2
-
dp2( ~
r'ia )
- 6 - -
O¢(r';x
-~o-~
- - +
~.¢)
"
(7)
We have 0¢/0a > 0, thus for Lemma 2 ~ will be negative definite if the condition,
-- (6/1)r'a ~- 2 Re [-r'/2l) (I -- 6G) (i¢oi - a)-'a-] > 0
(8)
is satisfied. Taking advantage of the identity: -- 6(iwI - G)-~(io~I - G) + 61 = O. It is possible after a few computations to rewrite this last condition in the form Re (1 - 6i~)F(io~) < 0
(9)
for all ~, where F(i~) is given by Eq. 10. Under such conditions, the function is then negative definite with respect to (x = 0, z = 0). The application of Lemma 1) then completes the proof. Recalling the transformation (4), this theorem then assures that the equilibrium point (2) is asymptotically stable for system (1), with the region of attraction being the region ~ > O. Of course it is not true that every solution of system (1) other than (2) approaches (3) as t --~ ~ : in fact, it is possible to prove (6) that there is at least one trajectory joining (2) with ~. The preceding theorem then assures that such a trajectory lies in the v _< 0 region and thus has no physical meaning. For the given system, the original Popov criterion is, (1, 5, 8)
(E/1) (~ - r'G-la) + (l/l) Re (1 + fli~)[r'G-i(i~I - G)-la~ > O.
(10)
It is an open question to investigate the relationships between these two conditions and to decide whi(~h particular problem gives the larger region of stability.
Vol. 284, No. 5, November 1967
281
,S. Albertoni, A. Cellina, G. P. Sze~o
~i'@/ICP.8 (I) 8. I.efl~etm, "Stability of Nonlinear Control Systems," New York, Academic Press, 1965. (2) W. IMran and K. Meyer, "Effect of Delayed Neutrons on the Stability of a Nuclear Power Reactor," N.S.E., Vol. 24, No. 4, p. 356, 1966. (3) S. Albertoni, A. Cellina and G. P. S~go, "Criteri di stabilitS, di reattori nucleari," Milano, Italy, Tamburini Editore, 1966. (4) G. P. Szego, "New Tlieorema on Stability and Attraction and their Application to a Control Problem," From Stability Problems of Solutions of Differential Equations, Editori "Oderisi" Gubbio (Italy), pp. 107-175, 1966. (5) Y. H. Ku and H. T. Chieh, "Extension of Popov's Theorcm for Stability of Nonlinear Control Systems," Jour. Frank. Inst., Vol. 279, No. 6, pp. 401-416, 1965. (6) N. P. Bhatia and G. P. Szego, "Dynamical System; Stability Theory and Applications," Lecture Notes on Mathematics, Springer-Verlag, (In print). (7) M. A. Aizerman and F. R. Gantmacher, "Absolute Stability of Regulator Systems," Sail Franeisto, Calif., Holden-Day, Inc. 1.964. (8) Y. H. Ku and H. T. Chieh, "New Theories oil Absolute Stability of Non-Autonomous Noalincar Control Systems," IEEE Itder~at. (YoJw. Rcc., Vol. 14, Pt. 7, pp. 260-271, 1966.
282
Journal of The Franklin Institute