E-Journal of Advanced Maintenance Vol.9-2 (2017) 132-137
Japan Society of Maintenology

Reactivity Measurement Based on a New Finite-Time Convergent 2-Order
Differentiator
Zhe Dong1,*1

Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced
Nuclear Energy Technology, Tsinghua University, Beijing 100084, China
ABSTRACTReactivity measurement is crucial in monitoring nuclear reactors. Proper estimation of neutron flux derivative is
one of the key to improve the reactivity estimation based on inverse kinetics (IK) method. Signal differentiation
is widely used in system monitoring and control. The key problem in differentiator design is how to obtain asatisfactory tradeoff between differentiation exactness and robustness to uncertainties and noises. By proposing
a new finite-time stabilizer for 2-order integrator chain, a new finite-time-convergent 2-order differentiator withbounded estimation error is developed. This differentiator is then applied to reactivity measurement throughsolving the IPK equation. Numerical simulation results not only verify the theoretical results but also show its
satisfactory performance.
KEYWORDS

ARTICLE INFORMATION

Reactivity measurement, differentiator, finite-time stability

Article history:
Received 7 November 2016
Accepted 27 April 2017

1.

Introduction

Reactivity estimation is crucial for the nuclear reactor monitoring and operational safety. Inverse
point kinetics (IPK) method is an estimation approach that is widely implemented in the practicalengineering [1]. For IPK method, the reactivity estimation is made continuously by solving the IPK
equation which is a differential-integral equation containing both differentiation of the neutron fluxand integration of the concentrations of the delayed neutron precursors. Some analog [2] and digital[3-7] reactivity meters based upon the IPK method were developed and implemented practically. Oneof the key aspects of applying the IPK method is to estimate the derivative of the neutron flux. The
key aspect of applying the IPK method is to estimate the derivative of the neutron flux. Improperdifferentiation approach would amplify the noises, and leads to unacceptable reactivity estimation.
Therefore, it is necessary to study the signal differentiation methods. In order to improve the precisionof signal differentiation in solving the IPK equation, the Lagrange method [8] are introduced forhigher performance of numerical differentiation. In [9], a sliding mode differentiator given in [10] wasapplied for the reactivity measurement, which suffers mainly from the inherent chattering effect. In
this paper, a novel finite-time stabilizer of 2-order integrator chain is proposed, which guarantees thatthe estimation error converges to a bounded set in a finite-time. This differentiator is applied toestimate the reactivity of nuclear reactors, and numerical simulation show the feasibility andsatisfactory performance of this newly-built 2-order differentiator.

2. A Finite-Time 2-Order Differentiator
Consider the global finite-time stabilization problem of the 2-order integrator chain given by x1  x2 , x2  u.
*(1)

Corresponding author, E-mail: dongzhe@mail.tsinghua.edu.cn

ISSN-1883-9894/10 © 2010 – JSM and the authors. All rights reserved.

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F. Lastname, et al./ Title of the Paper

where x=[x1, x2]T ∈R2 and u∈R.
Consider coordinate transformationTξ  ξ  x   1  2    x1
T1
1x2 1 q  r1 1 q x1  ,(2)where r1 is a positive constant, and 0≤q=2m/p<1/2 with both p and m being positive odd integers.
Theorem. Suppose that signal v(t)∈C2 is a continuous 2-order derivable signal withv  i   hi , i  1, 2 ,(3)where hi (i=1,2) are both positive constants. Consider system (1) with input u given byu   2 r2 21 2 q  x (4)where x   x1  v  t   x2  ,ξ2 is given by transformation (2), ε∈(0,1) is the perturbation parameter
T2q

1

r2  2 2  q   2q 1  2q  r11 q 2 q

1 q
1 q
1  q q 1
1  q 1 q q 1
 2 1  q   1 q r1  2 1  2q 1  q   1 q ,
1 q
1 q

(5)

r1=σ+2, and σ is a given strictly positive constant. Then, there ρ>0 so thatxi  v  i 1  O  1 1iq  , i  1, 2 ,(6)for t>εΓ, where Γ is a strictly positive bounded scalar.
In order to prove this theorem. Some definitions and lemmas are introduced as follows.
Consider autonomous nonlinear systemx  f  x  ,f  O   O,x  Rn ,(7)where f : U0→Rn is continuous on an open neighborhood U0 of the origin such that the solution ofsystem (7) is unique in the forward time.
Definition 1 [9]. The equilibrium point x=O of system (7) is called finite-time stable if it is
Lyapunov stable and finite-time convergent in a neighborhood U⊂U0 of the origin, where thefinite-time convergence means that there is a function T: U/{O} ↦(t0,∞) such that, for any x0∈U/{O},the solution x(t) to system (7) with x(t0)=x0 as the initial condition is defined, x(t)∈ U/{O} fort∈[t0,T(x0)), andlim x  t   Ot T  x0 

where T(x0) is called the settling time with respect to x0. If U = U0= Rn, then the origin is global
finite-time stable.
Definition 2. The nonlinear control systemx  f  x   g  x  u,x  Rn, u Rm ,(8)is called finite-time stabilizable via continuous state-feedback if there exists a continuous feedback
law u=u(x) such that the equilibrium of the closed-loop system is finite-time stable.
The following Lemma 1 gives a sufficient condition for the origin of system (8) is globally
finite-time stable.
Lemma 1 [9]. Consider autonomous system (7). If there is a real scalar α∈(0,1), a positive scalar
c and a C1 radially unbounded Lyapunov function V(x) of the system such thatV  x   cV   x   0(9)holds along the trajectories of system (7) starting from any x0, then the origin point is a global
finite-time stable equilibrium. Furthermore, the settling time with respect to x0 satisfies

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F. Lastname, et al./ Title of the PaperT  x0  1V 1  x0  ,c 1   (10)where x0= x(0)∈Rn.
Some lemmas that will be used in the sequel is introduced as follows.
Lemma 2 [8]. For any real numbers xi, i=1,…,n and 0<b≤ 1, the following inequality holds: x  x   xb1nb1b   xn .

(11)

If b=m/p≤ 1, where both m>0 and p>0 are odd integers, thenbx b  y b  21b x  y ,(12)where x, y are positive scalars.
Lemma 3 [8]. Let c, d be two positive real constants and γ(x, y)>0 be a real-valued function,
where x, y are positive scalars. Then, inequalitycdcdc  x, y  x
 d   c d  x, y  yc
dx y .
(13)cd
is well satisfied.
Proof of Theorem: Design the control input u of system (1) asu   r2 21 2 q  x Choose the Lyapunov function for system (1) as(14)2V  x   Wk  xk  ,(15)k 1wherexk   x1  xk  , k  1, 2,T(16)1 kq 1 k11 q
Wk  xk   Wk k 1 , xk    1 k 1q  s rk 1k 1  ds
 rk 1k 1
r0  0  0 .
From equation (17),1 kq
Wk  xk    kxk  rk 1 k11 k 1 q  2 k 1 q  k2  q ,xknV  x    2 k 1q  k2  q .

(17)(18)(19)(20)k 1

Differentiate function W1 along the trajectory of model (1),W1  r112  11 q  x2  r111 q   r112  2q 1

1 q

2

1 q
1 q

1 q
1  q 1 q q 1  2 1  q   1 q  22
   r1     (21)1 qSimilarly, based on Lemmas 2 and 3, differentiate W2 along the trajectory given by (1) and (14)1 qW2  r2 22  2q 1  2q  r1  2121 r2 22  2q 1  2q  r1  2

1 q

 2  r111 q   r2  2q 1  2q  r1   22  2q 1  2q  r1r1  2

1 q

1

1 q

1 q 
1  q 1 q 1 q q 1   r2  2q 1  2q  r1   22  12   r1  2 1  2q 1  q   1 q   22
1  qBased on inequalities (21) and (22), we then have1 q1342 q(22)

F. Lastname, et al./ Title of the PaperV  x   W1 1   W2 1 , x2    r1  2  12  r2  2q 1  2q  r1 
1 q
1 q 1  q 1 q q 1
1  q 1 q 1 q q 1  2 1  q   1 q  r1  2 1  2q 1  q   1 q   22 .
1 q
1 q
1 q

1 q

2q

(23)

From inequality (23), it is feasible to set
r1    2 ,2q
2 q

1 q
1 q

(24)

1 q
1 q

1 q
1 q  2q 1 1  q   

1 q
1 q
where σ is a given positive constant, and then from Lemma 2,r2  2

  2q 1  2q  r1 

1 q
1 q

2 q
1 q
1r

 2q 1 1  2q 1  q  1 q
1 q,(25)

22q
2

 2
 2 qV  x     12  2 2  q  22      2 k 1 q  k 2  q     V 2  q  x  .
(26)

1k


Based on Lemma 1, the closed-loop system formed by integrator chain (1) and the feedback law
given by (14), (2), and (24) is globally finite-time stable.
Define the observation error e asTTe   e1 e2    x1  v x2  v  ,(27)and define z   z1 z2 T ,(28)k 1
 zk   ek , k  1, 2.
From (1), (27) and (28), observation error e satisfies
 dz1
 d  z2 ,2
 dz2  r  1 2 q  z  v   d v ,(29)2 2 dd 2
wheret ,(30)

Tdv Tv  0   0  v  .
dChoose the Lyapunov function of system (29) as(31)2V  z   Wk  z  ,(32)k 1where function Wk is defined by (17), and k=1, 2.
By differentiating V(z) along the trajectory of system (29), and based on Lemma 2 and inequality(26), we can derive thatdV  z  dV  z 1 2 q
1 2 q1 2 q 2  z 2  z  v   2  z v  2  r2 22 q  2  z dd v  0

 2

22 k 1 q
2 q

2 k 121 kqk k2 z   M h2kk 1

2k
1 kq

(33)

where

M1  1  2q  22 q 1 2 q  2 q 1 2 q M 2  1  2q  2

1 2 q1 2 q2 q 1 2 q 4q
 2  q 1 2 q  1 2 q

1 2 q r2  n  11  nq  22 q 1  12 q ,135

1 2 q1 2 q1 2 q1  2q 1 2 q .

(34)(35)

F. Lastname, et al./ Title of the Paper

8x 10-4x 10-4

Simulated
Newly-built, ε = 0.01
Newly-built, ε = 0.1
Classical, τ = 0.02

86

64Reactivity ρReactivity ρ42
02
0-2-2-4-4-6
224.95225

time/s225.05299.95225.1

300300.05time/s300.1

300.15

Fig. 1. Simulation results: τ is the inertial of classical differentiator

For the initial point z0 in the outside of bounded set
2 k 1 q2k2 2
 2 2 q 2
21 kq  z  Rk  z    M k 2v  i  1 kq  ,4 k 1k 1and from (33), we have
2dV  z  2 q  V z ,d4which means that state-vector z is in set Ξ if t>εΓ with(36)

(37)

q

8  4q 2  q(38)
0 V  z0  .
qFurthermore, from coordinate transformation (2) there exist positive constants λk (k=1, 2) so that
1
1 kq

1  k 1 q

ek  k  hk 1 kq , k  1, 2 ,which means that relationship (6) can be well satisfied. This completes the proof of Theorem.
3.

(39)

Application to Reactivity Measurement with Simulation Results

The reactivity estimation is given by the following inverse kinetic equation d ln  nr dt

6

ite i t  nr0  i  ei nr   d  ,t0
i 1 n (40)rwhere nr is normalized neutron flux with initial value nr0, Λ is the prompt neutron lifetime, λi and βi isrespectively the precursor decay constant and fraction of the ith group delayed neutron. From (40),the derivative of ln(nr) is the key in reactivity estimation.
In order for verification the newly-built differentiator in reactivity measurement. Consider thefollowing case for simulation. Initially, the reactor is steady in 10% full power (FP), and at 200s, asquare wave of reactivity with amplitude of 15%β and period of 50s is induced periodically. The newfinite-time convergent 2-order differentiator and the classical 1-order differentiator are used. Here, thetransfer function of the 1-order classical differentiator isGD  s  s, s 1(41)and τ is the inertial constant. In this simulation, parameters σ and q of 2-order finite-time differentiator

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F. Lastname, et al./ Title of the Paper

given by (1) and (4) are chosen to be σ=10 and q=2/5, and different values of ε are adopted. The
inertial constant τ of classical differentiator (41) is chosen to be τ=0.02, which has no relationshipwith the parameters of newly-built finite-time differentiator. The simulation results with comparison
between 2-order finite-time differentiator and classical differentiator are shown in Fig. 1.
From Fig. 1, the reactivity estimation precision is higher if ε is smaller, which is in accordancewith relationship (6), and thus verifies the theoretic results given by Theorem. Comparing to classicaldifferentiator, thanks to nonlinear feedback law (14), the newly-built differentiator composed by (1)and (14) guarantees higher transient and steady performance.
4.

Conclusion

Motivated by importance of signal derivatives in providing high performance reactivity
measurement, a new finite-time convergent 2-order differentiator is proposed, whose key idea is totransfer the problem of differentiator to the problem of finite-time stabilization of a 2-order integrator
chain. It is analyzed theoretically that the estimation error of both the signal and its derivative
converges to a bounded set in a finite-time. This novel differentiator is applied to solving the IPK
equation for reactivity measurement. Numerical simulation not only verifies the correctness of thetheoretical results but also shows the satisfactory performance of this differentiator. One of the future
works is to extend this 2-order differentiator to high-order version.
Acknowledgement

This work was supported in part by National S&T Major Project, and in part by Natural Science
Foundation of China (NSFC) under Grant 61374045.
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