# Nonlinear Adaptive Dynamic Output-Feedback Power-Level Control of Nuclear Heating Reactors

Authors: Zhe Dong
Journal: Science and Technology of Nuclear Installations
 (2013)

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## Abstract

Due to the high safety performance of small nuclear reactors, there is a promising future for small reactors. Nuclear heating reactor (NHR) is a small reactor that has many advanced safety features such as the integrated arrangement, natural circulation at any power levels, self-pressurization, hydraulic control rod driving, and passive residual heating removing and can be applied to the fields of district heating, seawater desalination, and electricity production. Since the NHR dynamics has strong nonlinearity and uncertainty, it is meaningful to develop the nonlinear adaptive power-level control technique. From the idea of physically based control design method, a novel nonlinear adaptive power-level control is given for the NHR in this paper. It is theoretically proved that this newly built controller does not only provide globally asymptotic closed-loop stability but is also adaptive to the system uncertainty. Numerical simulation results show the feasibility of this controller and the relationship between the performance and controller parameters.


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## Body

## 1. Introduction

Due to the serious climate and environment problems such as global warming that caused by those greenhouse gases emitted from burning fossil fuels, it is significant for people to develop clean energy. Nuclear energy is one of the most rapidly developing clean energy, which gives the current rebirth of nuclear energy industry. After the severe Fukushima nuclear accident, much more attention has to be drawn on safety issues of nuclear plants. In contrast to those large nuclear reactors, small reactors usually has low power density which leads to higher safety performance and can be built near big cities for district heating, seawater desalination, and electricity production. Nuclear heating reactor (NHR) is a kind of small light water reactor, and the design of the NHR in China started since early 1980s. The first NHR of China, that is, the 5 MWth test nuclear heating reactor (NHR-5) began to be built in the Institute of Nuclear and New Energy Technology (INET) of Tsinghua University in March 1986 and reached its full power operation on December 16, 1989 [1–4]. Based on NHR-5, INET also designed the 200 MWth nuclear heating reactor NHR-200 which inherits many advanced safety characteristics from NHR-5 such as integrated arrangement, natural circulation at any operating power levels, self-pressurized performance, hydraulic control rod driver, and passive residual heat removing [5]. Both structure and cross-section of the NHR-200 are shown in Figure 1. As an improvement of the NHR-5, the NHR-200 can be applied to not only the areas of district heating, system [5] and seawater desalination [6] but also small-scale power generation for those consumers which are isolated from the central power grids. INET also gave some results on the interconnection between NHR and some desalination processes [7, 8]. Since the NHR can be used for electricity production, district heating, and seawater desalination, the load of the NHR may vary frequently in a wide range, and its power controller should provide the load-following performance.Structure and cross-section of the NHR-200: 1: primary heating exchanger, 2: riser, 3: biological shield, 4: containment, 5: pressure vessel, 6: core, 7: fuel elements, 8: control rods.
(a)
(b)The basic principle of power-level control is generating the insertion and withdrawal speed of control rods in order to regulate power output at a demand value using the average temperature deviation of the coolant inside the reactor core and power mismatch of the reactor. Power-level control is one of the most crucial techniques guaranteeing safe, stable, and efficient operation. With the development of high performance industrial microprocessors, it is now possible to implement advanced control laws for improving dynamic performance of nuclear reactors. Edwards designed a linear power-level control strategy named state-feedback assisted classical controller (SFAC) [9] which modifies reference signal of an embedded classical output feedback controller by state feedback and is easy to be implemented since it leaves the current classical feedback loop in place. Then, the technique of linear quadratic Gaussian regulation with the loop transfer recovery (LQG/LTR) was then applied to strengthen the robustness of the SFAC [10, 11]. Since nuclear reactors are essentially complex and nonlinear dynamic systems, it is necessary and meaningful to develop nonlinear power-level control strategy for guaranteeing the load following performance. Shtessel gave a nonlinear power-level control composed of a static state-feedback sliding mode control and a sliding mode state-observer for space nuclear reactor TOPAZ II [12]. Dong presented a dynamic output feedback dissipation controller [13] based on both backstepping technique [14] and dissipation-based high gain filter (DHGF) [15, 16]. Though these nonlinear controllers provide both globally asymptotic closed-loop stability and load-following performance, they are too complicated to be implemented practically. Nowadays, designing effective control by fully using the good features of natural system dynamics, that is, the physically based control method [17–19], has already become a hot spot of modern control theory. Since every nuclear reactor is essentially a thermodynamic system with the nuclear fission reaction whose evolution trend is given by the differences of both temperature and concentration but not energy, the existing physically based method cannot be applied directly. Based upon defining the shifted-ectropy for the modular high temperature gas-cooled reactor (MHTGR) [20], Dong gave a physically based power-level control for the MHTGR [21] which provides globally asymptotic closed-loop stability and takes a simple form. Since every nuclear reactor is actually an uncertain dynamic system, it is very meaningful to design the adaptive power-level control law. Actually, adaptive output-feedback control of nonlinear systems is also a hot area in control theory and has also been applied to some practical systems. Fu presented a dynamic output feedback adaptive control law for nonlinear systems in generalized output feedback canonical form based on the extended backstepping approach [22]. Yang et al. gave an adaptive robust output feedback nonlinear control for magnetic levitation system by merging a disturbance observer into the K-filter-based adaptive control design technique [23]. Mackunis et al. developed an adaptive output dynamic inversion control for an unmanned aerial vehicle (UAV) which guarantees global asymptotic tracking performance [24]. Xu and Huang gave a global robust adaptive output feedback control for a class of uncertain nonlinear systems based on the internal mode design approach [25]. Yao et al. proposed an adaptive precision motion control for a high-speed industrial gantry by the desired compensation adaptive robust control (DCARC) approach [26]. Since nuclear energy systems have their own dynamic features, the above adaptive output feedback control design methods cannot fully use the subdynamics of nuclear energy systems that is beneficial to stabilization and may lead to control laws with complicated forms that are hard to implement. Therefore, design adaptive power-level control strategies by the use of the good parts of nuclear energy system dynamics is very meaningful. Park and Cho gave a PI power-level control law with feedback gains adjusted by an adaptive law [27]. Arab-Alibeik and Setayeshi designed an adaptive power-level control by using a feedforward neural network [28] for the pressurized water reactor (PWR), and the neural network is trained online. Dong proposed a novel nonlinear adaptive power-level controller which provides not only the globally asymptotic closed-loop stability but also the adaption capability to system uncertainties [29].With comparison to the common PWRs, the feature of natural circulation at any power-levels strengthen the nonlinearity of the NHR dynamics. Moreover, there must be modeling uncertainty and exterior disturbance. Thus, to guarantee the load following performance, it is quite meaningful to develop nonlinear adaptive power-level control strategy for the NHR. In this paper, a dynamic output-feedback nonlinear adaptive power-level control law is presented for the NHR through the physically based approach. This new power-level control guarantees load-following performance through providing both the globally asymptotic closed-loop stability and the adaption capability to slowly varying uncertainty. This newly built control is then applied to the power-level regulation of a NHR, and numerical simulation results show both its feasibility and the relationship between its control performance and values of controller parameters.


## 2. Problem Formulation

In this section, the nonlinear state-space model for designing the power-level control strategy is firstly given, and then the theoretic problem corresponding to the control design is established.


### 2.1. Nonlinear State-Space Model

The reactor dynamics adopted here for power-level control design is the point kinetics with one equivalent delayed neutron group and the reactivity feedback determined by the variations of average temperatures of the fuel and coolant, which is given as follows:(1)n˙r=1Λ[(ρr-β)nr+βcr+αfnr(Tf-Tf,m)+αcnr(Tcav-Tcav,m)],c˙r=λ(nr-cr),T˙f=1μf[-Ω(Tf-Tcav)+P0nr],T˙cav=1μc[Ω(Tf-Tcav)-2M(Tcav-Tcin)],ρ˙r=Grzr,
where nr is the relative nuclear power, cr is the relative concentration of delayed neutron precursor, β is the fraction of delayed neutrons, Λ is the effective prompt neutron lifetime, λ is the effective radioactive decay constant of delayed neutron precursor, αf and αc are, respectively, the reactivity coefficients of the fuel and coolant temperatures, Tf is the average fuel temperature, Tcav and Tcin are, respectively, the average temperature of the coolant inside the core and the temperature of the coolant entering the core, Tcav,m and Tcin,m are, respectively, the initial equilibrium values of Tcav and Tcin, Ω is the heat transfer coefficient between fuel and coolant, M is the mass flow rate times heat capacity of the coolant, P0 is the rated power level, ρr is the reactivity due to the control rods, μf is the total heat capacity of the fuel, μc is the total heat capacity of the reactor coolant, Gr is the total reactivity worth of control rods, and zr is the control input, that is, the speed signal of control rods.Suppose thatnr0, cr0, Tf0, Tcav0, Tcin0, and  ρr0 are, respectively, the steady values of nr, cr, Tf, Tcav, Tcin, and ρr, which satisfies
(2)n˙r0=c˙r0=T˙f0=T˙cav0=T˙cin0=ρ˙r0=0.From (1) and (2), we can easily see that these steady values should satisfy
(3)nr0=cr0,ρr0=-αf(Tf0-Tf,m)-αc(Tcav0-Tcav,m),P0nr0=Ω(Tf0-Tcav0)=2M(Tcav0-Tcin0).Define the deviations between the actual and the steady values ofnr, cr, Tf, Tcav, Tcin, and ρr as
(4)δnr=nr-nr0,δcr=cr-cr0,δTf=Tf-Tf0,δTcav=Tcav-Tcav0,δρr=ρr-ρr0,δTcin=Tcin-Tcin0.Then, based upon (3), dynamic model (1) can be rewritten as
(5)δn˙r=1Λ[-β(δnr-δcr)+nr(αfδTf+αcδTcav+δρr)],δc˙r=λ(δnr-δcr),δT˙f=1μf[-Ω(δTf-δTcav)+P0δnr],δT˙cav=1μc[Ω(δTf-δTcav)-2M(δTcav-δTcin)],δρ˙r=Grzr.It is worth to be noted here that the physical parameters,β, Λ, Ω, M, μf, and μc cannot be obtained accurately, which results in parameter uncertainty. Seeing δTdin as an exterior disturbance also leads to uncertainty. Furthermore, there must exist uncertainty which comes from unmodeled dynamics. Based on this discussion, the nonlinear state-space model is
(6)x˙=f(x)+g(x)ξ+σ,ξ˙=u,
where
(7)x=[δnrδcrδTfδTcav]T,(8)ξ=δρr,(9)u=Grzr,(10)σ=[σ1σ2σ3σ4]T,(11)f(x)=[-βΛ(x1-x2)+nr0+x1Λ(αfx3+αcx4)λ(x1-x2)-Ωμf(x3-x4)+P0μfx1P0μfΩμc(x3-x4)-2Mμcx4],(12)g(x)=[nr0+x1Λ000]T.
Here σ∈R4 is a norm-bounded vector denoting the uncertainty. Both x and ξ are the system state, and u is the control input.


### 2.2. Formulation of Theoretic Problem

Ifx1≡0, then from the first two equations of model (6), we have
(13)βx2+nr0(αfx3+αcx4+ξ)+σ1≡0,(14)x˙2=-λx2+σ2.It is easy to see from (14) that
(15)x2=x20e-λt+∫t0te-λ(t-s)σ2(s)ds,
where x20 is the initial value of x2. Substitute (15) to (13),
(16)βx20e-λt+β∫t0te-λ(t-s)σ2(s)ds+nr0(αfx3+αcx4)+(nr0ξ+σ1)≡0.
Since values of real scalars σ1, σ2, x20, ξ are all arbitrary, (16) is satisfied if and only if
(17)x2=σ2=αfx3+αcx4=nr0ξ+σ1≡0.Moreover, from (17) and the second two equations of model (6), we can easily derive that
(18)-αfΩμf(x3-x4)+αfσ3+αcΩμc(x3-x4)-2Mαcμcx4+αcσ4=αfx˙3+αcx˙4≡0,
that is
(19)-[Ω(αcμc-αfμf)(1+αcαf)+2Mαcμc]x4+αfσ3+αcσ4≡0.Since bothσ3 and σ4 are arbitrary real scalars, it is clear that (19) holds if and only if
(20)x4=σ3=σ4≡0.Based on (17) and (20), we can see that if x1≡0, then
(21)σ2=σ3=σ4≡0.Thus, from (21),
(22)nr=nr0+x1>0,cr=cr0+x2=nr0+x2>0,
it is reasonable to let σ be
(23)σ=[(nr0+x1)θ1x1(nr0+x2)θ2x1θ3x1θ4]T,
where
(24)|θi|≤Mi,i=1,…,4,
and Mi(i=1,2,3,4) are all given positive scalars.There might be other forms ofσ that satisfy the property. However, (23) is reasonable with simple form. Based upon the above derivation and analysis, the theoretic problem to be solved in this paper can be formed as follows.Problem 1.
Consider uncertain nonlinear system (6). How to design control input u so that the closed-loop system is not only globally asymptotically stable but also adaptive to uncertainty σ that takes the form of (23)?


## 2.1. Nonlinear State-Space Model

The reactor dynamics adopted here for power-level control design is the point kinetics with one equivalent delayed neutron group and the reactivity feedback determined by the variations of average temperatures of the fuel and coolant, which is given as follows:(1)n˙r=1Λ[(ρr-β)nr+βcr+αfnr(Tf-Tf,m)+αcnr(Tcav-Tcav,m)],c˙r=λ(nr-cr),T˙f=1μf[-Ω(Tf-Tcav)+P0nr],T˙cav=1μc[Ω(Tf-Tcav)-2M(Tcav-Tcin)],ρ˙r=Grzr,
where nr is the relative nuclear power, cr is the relative concentration of delayed neutron precursor, β is the fraction of delayed neutrons, Λ is the effective prompt neutron lifetime, λ is the effective radioactive decay constant of delayed neutron precursor, αf and αc are, respectively, the reactivity coefficients of the fuel and coolant temperatures, Tf is the average fuel temperature, Tcav and Tcin are, respectively, the average temperature of the coolant inside the core and the temperature of the coolant entering the core, Tcav,m and Tcin,m are, respectively, the initial equilibrium values of Tcav and Tcin, Ω is the heat transfer coefficient between fuel and coolant, M is the mass flow rate times heat capacity of the coolant, P0 is the rated power level, ρr is the reactivity due to the control rods, μf is the total heat capacity of the fuel, μc is the total heat capacity of the reactor coolant, Gr is the total reactivity worth of control rods, and zr is the control input, that is, the speed signal of control rods.Suppose thatnr0, cr0, Tf0, Tcav0, Tcin0, and  ρr0 are, respectively, the steady values of nr, cr, Tf, Tcav, Tcin, and ρr, which satisfies
(2)n˙r0=c˙r0=T˙f0=T˙cav0=T˙cin0=ρ˙r0=0.From (1) and (2), we can easily see that these steady values should satisfy
(3)nr0=cr0,ρr0=-αf(Tf0-Tf,m)-αc(Tcav0-Tcav,m),P0nr0=Ω(Tf0-Tcav0)=2M(Tcav0-Tcin0).Define the deviations between the actual and the steady values ofnr, cr, Tf, Tcav, Tcin, and ρr as
(4)δnr=nr-nr0,δcr=cr-cr0,δTf=Tf-Tf0,δTcav=Tcav-Tcav0,δρr=ρr-ρr0,δTcin=Tcin-Tcin0.Then, based upon (3), dynamic model (1) can be rewritten as
(5)δn˙r=1Λ[-β(δnr-δcr)+nr(αfδTf+αcδTcav+δρr)],δc˙r=λ(δnr-δcr),δT˙f=1μf[-Ω(δTf-δTcav)+P0δnr],δT˙cav=1μc[Ω(δTf-δTcav)-2M(δTcav-δTcin)],δρ˙r=Grzr.It is worth to be noted here that the physical parameters,β, Λ, Ω, M, μf, and μc cannot be obtained accurately, which results in parameter uncertainty. Seeing δTdin as an exterior disturbance also leads to uncertainty. Furthermore, there must exist uncertainty which comes from unmodeled dynamics. Based on this discussion, the nonlinear state-space model is
(6)x˙=f(x)+g(x)ξ+σ,ξ˙=u,
where
(7)x=[δnrδcrδTfδTcav]T,(8)ξ=δρr,(9)u=Grzr,(10)σ=[σ1σ2σ3σ4]T,(11)f(x)=[-βΛ(x1-x2)+nr0+x1Λ(αfx3+αcx4)λ(x1-x2)-Ωμf(x3-x4)+P0μfx1P0μfΩμc(x3-x4)-2Mμcx4],(12)g(x)=[nr0+x1Λ000]T.
Here σ∈R4 is a norm-bounded vector denoting the uncertainty. Both x and ξ are the system state, and u is the control input.


## 2.2. Formulation of Theoretic Problem

Ifx1≡0, then from the first two equations of model (6), we have
(13)βx2+nr0(αfx3+αcx4+ξ)+σ1≡0,(14)x˙2=-λx2+σ2.It is easy to see from (14) that
(15)x2=x20e-λt+∫t0te-λ(t-s)σ2(s)ds,
where x20 is the initial value of x2. Substitute (15) to (13),
(16)βx20e-λt+β∫t0te-λ(t-s)σ2(s)ds+nr0(αfx3+αcx4)+(nr0ξ+σ1)≡0.
Since values of real scalars σ1, σ2, x20, ξ are all arbitrary, (16) is satisfied if and only if
(17)x2=σ2=αfx3+αcx4=nr0ξ+σ1≡0.Moreover, from (17) and the second two equations of model (6), we can easily derive that
(18)-αfΩμf(x3-x4)+αfσ3+αcΩμc(x3-x4)-2Mαcμcx4+αcσ4=αfx˙3+αcx˙4≡0,
that is
(19)-[Ω(αcμc-αfμf)(1+αcαf)+2Mαcμc]x4+αfσ3+αcσ4≡0.Since bothσ3 and σ4 are arbitrary real scalars, it is clear that (19) holds if and only if
(20)x4=σ3=σ4≡0.Based on (17) and (20), we can see that if x1≡0, then
(21)σ2=σ3=σ4≡0.Thus, from (21),
(22)nr=nr0+x1>0,cr=cr0+x2=nr0+x2>0,
it is reasonable to let σ be
(23)σ=[(nr0+x1)θ1x1(nr0+x2)θ2x1θ3x1θ4]T,
where
(24)|θi|≤Mi,i=1,…,4,
and Mi(i=1,2,3,4) are all given positive scalars.There might be other forms ofσ that satisfy the property. However, (23) is reasonable with simple form. Based upon the above derivation and analysis, the theoretic problem to be solved in this paper can be formed as follows.Problem 1.
Consider uncertain nonlinear system (6). How to design control input u so that the closed-loop system is not only globally asymptotically stable but also adaptive to uncertainty σ that takes the form of (23)?


## 3. Nonlinear Adaptive Power-Level Control Design

In this part, Problem1 which is raised at the end of Section 2 will be solved step by step through establishing the nonlinear adaptive power-level control law for the NHR.


### 3.1. State-Feedback Control Design

Due to the idea of backstepping method for nonlinear control law design,ξ can be regarded as a virtual control, and the first step is to design its referenced value ξr that stabilizes subsystem
(25)x˙=f(x)+g(x)ξ+σ,
where vector-valued functions f, g, and σ are determined by (11), (12), and (23), respectively.Theorem2 gives a nonlinear adaptive state-feedback stabilizer for subsystem (25), which is the first main result of this paper.Theorem 2.
Consider subsystem (25), and assume that uncertain parameters θi(i=1,…,4) vary slowly, that is,
(26)θ˙≫O,
where
(27)θ=[θ1θ2θ3θ4]T.
Consider state-feedback adaptive virtual control law
(28)ξr=-knx1-kfx3-kcx4-θ^TΓω,
where positive scalars kn, kf, and kc are all feedback gains and θ^ is the estimation of  θ  satisfying
(29)θ^=[θ^1θ^2θ^3θ^4]T
and determined by adaptive law
(30)θ^˙=x1Q-1ω,(31)kc=Ωμc(M+Ω)μf(kf-αf)+αc,(32)ω=[1x2(1+ΩM)x3+ΩμcMμfx4ΩMx3+(1+ΩM)x4]T,(33)Γ=diag([Λβλkf′μf(1+Ω/M)P0kf′μc(1+Ω/M)P0]T),(34)kf′=kf-αf,(35)Q=diag([q1q2q3q4]),qi>0,i=1,…,4.
Adaptive virtual control strategy (28) guarantees globally asymptotic closed-loop stability for sub-system (25), if kf′ is a positive scalar.Proof.
Based upon the shifted-ectropy for nuclear reactors [19], the Lyapunov function of the neutron kinetics can be chosen as
(36)ζN(x1,x2)=Λ[1+x1nr0-ln(1+x1nr0)]+βλ[1+x2nr0-ln(1+x2nr0)].
It is clear that ζN is positive definite, and differentiation satisfies
(37)ζ˙N=-β(x1-x2)2nr0(nr0+x1)(nr0+x2)+x1nr0(αfx3+αcx4+ξr)+x1nr0(Λθ1+βλx2θ2).
Substitute (28) to (37), and we have
(38)ζ˙N=-β(x1-x2)2nr0(nr0+x1)(nr0+x2)-knnr0x12-x1nr0(kf′x3+kc′x4)-x1nr0ωTΓθ^+x1nr0(Λθ1+βλx2θ2),
where
(39)kc′=kc-αc.
Moreover, choose the Lyapunov function for the reactor thermal-hydraulic loop as(40)ζT(x3,x4)=12μfx32+12μcx42+Ω2Mμf(μfx3+μcx4)2,
where the first and second terms of ζT are, respectively, the shifted-ectropy of the fuel and the coolant, and the third term is the shifted-ectropy of the total thermal-hydraulic loop. Differentiate ζT along the trajectory given by (25), and we can derive that
(41)ζ˙T=-Ω(x3-x4)2-2Mx42+P0x1x3+x1(μfx3θ3+μcx4θ4)-ΩMμf×{[2Mμfx3x4-2Mμcx42+P0x1(μfx3+μcx4)]gg+x1(μfx3+μcx4)(μfθ3+μcθ4)x42}=-Ωx32-Ω(1+2MΩ+2μcμf)x42+P0x1[(1+ΩM)x3+ΩμcMμfx4]+μfx1[(1+ΩM)x3+ΩμcMμfx4]θ3+μcx1[(1+ΩμcMμf)x3+ΩMx4]θ4.
Now, select the Lyapunov function for subsystem (25) as
(42)V1(x)=nr0ζT(x1,x2)+kf′(1+Ω/M)P0ζN(x3,x4),
and we can easily see that V1 is positive definite if and only if kf′ is a positive scalar. Then, from (38) and (41), the differentiation of V1 along the trajectory given by (25) and (28) can be represented as
(43)V˙1(x)=nr0ζ˙T(x1,x2)+kf′(1+Ω/M)P0ζ˙N(x3,x4)=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]+kf′x1[x3+Ωμc(M+Ω)μfx4]-x1(kf′x3+kc′x4)-x1ωTΓθ^+x1(Λθ1+βλx2θ2)+kf′x1(1+Ω/M)P0×{μf[(1+ΩM)x3+ΩμcMμfx4]θ3dd+μc[ΩMx3+(1+ΩμcMμf)x4]θ4}.
From (31), (32), and (33), we have
(44)V˙1(x)=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0×[x32+(1+2MΩ+2μcμf)x42]+x1θ~TΓω,
where
(45)θ~=θ-θ^
is the estimation error of uncertainty vector θ.
Choose the Lyapunov function for the closed-loop system formed by subsystem (25), virtual controller (28), and adaptive law (30) as
(46)V2(x,θ~)=V1(x)+12θ~TQΓθ~=nr0ζT(x1,x2)+kf′(1+Ω/M)P0ζN(x3,x4)+12θ~TQΓθ~.
Under assumption (26), we can obtain that
(47)θ~˙=θ˙-θ^˙=-θ^˙=-x1Q-1ω.
Differentiate V2 along the trajectory given by (25), (28), and (30), and then we have
(48)V˙2(x,θ~)=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42],
from which we can see that adaptive virtual control (28) provides globally asymptotic closed-loop stability for subsystem (25), if kf′ is positive. This completes the proof of this theorem.Remark 3.
From adaptive law (29), we have
(49)θ^=Q-1∫t0tx1ωdτ,
where t and t0 are, respectively, the current and initial time. Substitute (49) to control law (28), and it is clear that
(50)ξr=-knx1-kfx3-kcx4-ωTΓQ-1∫t0tx1ωdτ.
Based upon (50), this nonlinear adaptive controller takes the form as a PI control, and the integration term here is just introduced to compensate the uncertainty.Remark 4.
From (46), positive scalars qi(i=1,…,4) is the weighting factors of θ~i in V2. Here, qi is larger and the influence of V1 to V2 is smaller, which may lead to worse control performance.Nonlinear adaptive stabilizer (28) determines the referenced value of ξ, that is, the reactivity that should be induced by the control rods. In the following part of this subsection, this stabilizer is used to design control input u for entire system (6). Design and performance analysis of control input u is summarized as Theorem 8, which is the second main result in this paper.Theorem 5.
Consider control input of entire system (6)
(51)u=-kξ1x1-kξ2eξ+ξ˙r,
where both scalars kξ1 and kξ2 are positive, ξr is just the virtual control given in Theorem 2, and
(52)eξ=ξ-ξr.
Then, control input (51) can guarantee the globally asymptotic closed-loop stability.Proof.
Choose the Lyapunov function for the closed-loop system formed by system (6) and control input (51) as
(53)V3(x,θ~,eξ)=V2(x,θ~)+12kξ1eξ2=nr0ζT(x1,x2)+kf′(1+Ω/M)P0ζN(x3,x4)+12θ~TQΓθ~+12kξ1eξ2.
DifferentiateV3 along the trajectory given by system (6), control input (51), virtual control (28), and adaptive law (30), and we can derive that
(54)V˙3=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]+x1eξ+eξkξ1(u-ξ˙r)=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]-kξ2kξ1eξ2.
From (54), state vector of entire system (6) converges to the origin asymptotically, which completes the proof of Theorem 5.


### 3.2. State-Observer Design

Although nonlinear adaptive control (51) guarantees the globally asymptotical convergence of the reactor state, the precondition is that state-vector x can be measured directly. Actually, x2 and x3, that is, variations of the relative concentration of delayed neutron precursor and the average fuel temperature, cannot be measured directly. Therefore, it is necessary to design a state-observer. Theorem 6 gives a bounded state-observer of system (6), which is the third main result.Theorem 6.
Suppose (26) is well satisfied, and moreover assume that
(55)cr=cr0+x2=nr0+x2≤C.
Consider the state-observer of system (6) that takes the form as
(56)x^˙=f(x^)+g(x^)ξ^+Koeo+Hoϑo,ξ^˙=u+koξ(nr0+x1)e1,
where x^=[x^1x^2x^3x^4]T∈R4 is the state-observation, ξ^ is the observation of ξ, and vector-valued functions f and g are determined by (11) and (12), respectively,(57)eo=[e1e4]T=[x1-x^1x4-x^4]T,(58)ϑo=[ϑ1ϑ4]T,(59)Ko=[kon+αfx^3+αcx^4+ξ^Λ2λ-P0μf[kof(nr0+x1)-1]-αcP0αfμfkof(nr0+x1)002Ωμfkoc]T,(60)Ho=[nr0+x1000000x1]T,(61)ϑ˙o=φ(x1,eo),(62)φ(eo,x1)=[(nr0+x1)e1x1e4]T,and positive scalars kon, kof, and koc are all observer gains. Observer (56) provides state-observation with bounded observing error that can be satisfactory small if both observer gains koN and koT are high enough.Proof.
Define the observer error vector as(63)e=[e1e2e3e4e5]=[xT-x^Tξ-ξ^]T.
From (6) and (56), it is clear that the observation error e satisfies
(64)Λe˙1=-β(e1-e2)+(nr0+x1)(αfe3+αce4+e5)+Λ(nr0+x1)ϑ~1-Λkone1,e˙2=λ(e1-e2)-2λe1+x1(nr0+x2)θ2,μfe˙3=P0e1-Ω(e3-e4)+P0[kof(nr0+x1)-1]e1-2Ωe4+μfx1θ3,μce˙4=Ω(e3-e4)-2Me4+μcx1ϑ~4-μckoce4+αcαfP0kof(nr0+x1)e1,e˙5=-koξ(nr0+x1)e1,
where
(65)ϑ~k=θk-ϑk,k=1,4.
Choose the Lyapunov function for observer error dynamics (64) as
(66)Ve1(e)=Λ2e12+β2λe22+|αf|2P0kof(μfe32+μce42)+12koξe52,
and by differentiating Ve1 along the trajectory given by (64), we can derive that
(67)V˙e1=-β(e12+e22)-Λkone12+(nr0+x1)(αfe1e3+αce1e4+e1e5)+Λ(nr0+x1)e1ϑ~1+βλ(nr0+x2)x1e2θ2+|αf|P0kof×[(e3+αcαfe4)-Ω(e32+e42)-2Me42-μde42+kofP0(nr0+x1)e1×(e3+αcαfe4)+μfx1e3θ3+μcx1e4ϑ~4]-(nr0+x1)e1e5=-β(e12+e22)-|αf|ΩP0kof[e32+(1+2MΩ)e42]-Λkone12-|αf|μckocP0kofe42+Λ(nr0+x1)e1ϑ~1+βλ(nr0+x2)x1e2θ2+|αf|x1P0kof(μfe3θ3+μce4ϑ~4).
Then, define the Lyapunov function of both observer (56) and parameter adaptive law (61) as
(68)Ve(e,ϑ~o)=Ve1(e)+12ϑ~oTΠϑ~o,
where
(69)ϑ~o=[θ1-ϑ1θ4-ϑ4]T,
and Π is a diagonal positive-definite matrix given by
(70)Π=diag([Λ|αc|μcP0kof]).
Moreover, since we have assumed that (26), it is clear that
(71)ϑ~˙o=-ϑ˙o=-φ(x1,eo).
Differentiate Ve along the trajectory given by observer (56) and adaptive law (71),
(72)V˙e=V˙e1+ϑ~oTΠϑ~˙o=-β(e12+e22)-|αf|ΩP0kof[e32+(1+2MΩ)e42]-Λkone12-|αf|μckocP0kofe42+Λ(nr0+x1)e1ϑ~1+βλ(nr0+x2)x1e2θ2+|αf|x1P0kof(μfe3θ3+μce4ϑ~4)-ϑ~oTΠφ(eo,x1)=-β(e12+e22)-|αf|ΩP0kof[e322+(1+2MΩ)e42]-Λkone12-|αf|μckocP0kofe42+βλ(nr0+x2)x1e2θ2+|αf|μfP0kofx1e3θ3≤-β(e12+e222)-|αf|ΩP0kof[e322+(1+2MΩ)e42]-β2[e2-1λ(nr0+x1)x1θ2]2-|αf|Ω2P0kof(e3-μfΩx1θ3)2+β2λ2(nr0+x2)2x12θ22+|αf|μf22ΩP0kofx12θ32-(Λkone12+|αf|μckocP0kofe42).
Based on both assumptions (26) and (55), it is clear from (72) that
(73)V˙e≤-eTMe+CMx12-(Λkone12+|αf|μckocP0kofe42),
where
(74)M=diag([ββ2|αf|Ω2P0kof|αf|ΩP0kof(1+2MΩ)0]),CM=β2λ2C2M22+|αf|μf22ΩP0kofM32.
From inequality (73), the observation error e enters to the ball given by
(75)Be={e∣eTMe≤CMx12-(Λkone12+|αf|μckocP0kofe42)}
from which we can see that if both kon and koc are larger, then the observation error is nearer to the origin. This completes the proof of Theorem 6.Remark 7.
From inequality (73), if
(76)limt→∞x1(t)=0,
then the observation error vector enters to the set
(77)Se={e∣e1=e2=e3=e4≡0}.
Further, from the last equation of observation error dynamics (64), e5 is a bounded constant.


### 3.3. Stability of Closed-Loop Stability

From Theorems2, 5, and 6, the nonlinear adaptive dynamic output-feedback power-level control strategy for the nuclear heating reactor can be written as
(78)u=-kξ1x1-kξ2eξ+ξ^˙r,ξ^r=-knx1-kfx^3-kcx4-θ^TΓω^,θ^˙=x1Q-1ω^,x^˙=f(x^)+g(x^)ξ^+Koeo+Hoϑo,ξ^˙=u+koξ(nr0+x1)e1,ϑ˙o=φ(x1,eo),
where
(79)ω^=[1x^2(1+ΩM)x^3+ΩμcMμfx4ΩMx^3+(1+ΩM)x4]T,
scalar kc is given by (31), matrices Γ, Q, Ko, and Ho are, respectively, given by (33), (35), (59), and (60), and vector-valued functions f, g, and φ are determined by (11), (12), and (62), respectively.In this section, we will study whether nonlinear adaptive control law (79) can provide the globally asymptotic stability of the reactor. The result for stability analysis of the closed-loop system formed by system (6) and control strategy (78) is summarized as Theorem 8 which is the fourth main result of this paper.Theorem 8.
Suppose that both (26) and (55) are well satisfied. Consider nonlinear adaptive dynamic output-feedback power-level control (78). If
(80)k-n=kn-β2λ2(1+C)2M22-kf2kofP02|αf|Ω×{1+μfP0(1-αfkf)[1+kof-1(1-kfαf)-1+Ωμf(M+Ω)μc(1-kfαf)-1](1-kfαf)-1}2M32
is a positive scalar, then the closed-loop system constituted by system (6) and control strategy (78) is globally asymptotically stable.Proof.
Similar to Theorems2, 5, and 6, Theorem 8 will also be proved by Lyapunov direct method. Select the Lyapunov function for the entire closed-loop system formed by (6) and (78) as
(81)V4(x,θ~,eξ,e,ϑ~o)=V3(x,θ~,eξ)+Ve(e,ϑ~o).
Differentiate V4 along the trajectory determined by both system model (6) and control (78),
(82)V˙4=V˙3(x,θ~,eξ)+V˙e(e,ϑ~o)=-knx12-β(x1-x2)2(nr0+x1)(nr0+x2)-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]+θ~TΓ(θ~˙+x1ω^)+βλθ2e2x1+[kf+kf′μf(1+Ω/M)P0(1+ΩM+ΩμcMμf)θ3]e3x1-kξ2kξ1eξ2-(β+Λkon)e12-βe22-|αf|ΩkofP0[e32+(1+2M+μckocΩ)e42]+βλ(nr0+x2)θ2e2x1+|αf|μfkofP0θ3e3x1=-β(x1-x2)2(nr0+x1)(nr0+x2)-kf′Ω(1+Ω/M)P0×[x32+(1+2MΩ+2μcμf)x42]-kξ2kξ1eξ2-(β+Λkon)e12-β2e22-|αf|ΩkofP0[e322+(1+2M+μckocΩ)e42]-β2(e2-1+nr0+x2λθ2x1)2-|αf|Ω2kofP0{e3-kof[kfP0|αf|Ω+μfΩ(1-kfαf)ddddddffffff×(1+kof-1(1-kfαf)-1ddddddffffffggghhh+Ωμc(M+Ω)μf)]gggggg×θ3x1(1-kfαf)}2-{kn-(1+nr0+x2)22λ2θ22-kofP02|αf|Ωgggig×[kf+kf′μfP0(1+kof-1(1-kfαf)-1dddkkkkkkkkkkk+Ωμc(M+Ω)μf)]2×θ32}x12≤-k-nx12-β(x1-x2)2(nr0+x1)(nr0+x2)-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]-kξ2kξ1eξ2-(β+Λkon)e12-β2e22-|αf|ΩkofP0[e322+(1+2M+μckocΩ)e42],
where kf′ is defined in (34).
It is clear from inequality (82) that if k-n is a positive scalar, then the state of the closed-loop system finally enters into the set
(83)Θ={x∈R4,e∈R5,eξ∈R∣llllllllxk=0,ek=0,eξ=0,k=1,…,4},
which means that the corresponding closed-loop system is globally asymptotically stable. This completes the proof of this theorem.


## 3.1. State-Feedback Control Design

Due to the idea of backstepping method for nonlinear control law design,ξ can be regarded as a virtual control, and the first step is to design its referenced value ξr that stabilizes subsystem
(25)x˙=f(x)+g(x)ξ+σ,
where vector-valued functions f, g, and σ are determined by (11), (12), and (23), respectively.Theorem2 gives a nonlinear adaptive state-feedback stabilizer for subsystem (25), which is the first main result of this paper.Theorem 2.
Consider subsystem (25), and assume that uncertain parameters θi(i=1,…,4) vary slowly, that is,
(26)θ˙≫O,
where
(27)θ=[θ1θ2θ3θ4]T.
Consider state-feedback adaptive virtual control law
(28)ξr=-knx1-kfx3-kcx4-θ^TΓω,
where positive scalars kn, kf, and kc are all feedback gains and θ^ is the estimation of  θ  satisfying
(29)θ^=[θ^1θ^2θ^3θ^4]T
and determined by adaptive law
(30)θ^˙=x1Q-1ω,(31)kc=Ωμc(M+Ω)μf(kf-αf)+αc,(32)ω=[1x2(1+ΩM)x3+ΩμcMμfx4ΩMx3+(1+ΩM)x4]T,(33)Γ=diag([Λβλkf′μf(1+Ω/M)P0kf′μc(1+Ω/M)P0]T),(34)kf′=kf-αf,(35)Q=diag([q1q2q3q4]),qi>0,i=1,…,4.
Adaptive virtual control strategy (28) guarantees globally asymptotic closed-loop stability for sub-system (25), if kf′ is a positive scalar.Proof.
Based upon the shifted-ectropy for nuclear reactors [19], the Lyapunov function of the neutron kinetics can be chosen as
(36)ζN(x1,x2)=Λ[1+x1nr0-ln(1+x1nr0)]+βλ[1+x2nr0-ln(1+x2nr0)].
It is clear that ζN is positive definite, and differentiation satisfies
(37)ζ˙N=-β(x1-x2)2nr0(nr0+x1)(nr0+x2)+x1nr0(αfx3+αcx4+ξr)+x1nr0(Λθ1+βλx2θ2).
Substitute (28) to (37), and we have
(38)ζ˙N=-β(x1-x2)2nr0(nr0+x1)(nr0+x2)-knnr0x12-x1nr0(kf′x3+kc′x4)-x1nr0ωTΓθ^+x1nr0(Λθ1+βλx2θ2),
where
(39)kc′=kc-αc.
Moreover, choose the Lyapunov function for the reactor thermal-hydraulic loop as(40)ζT(x3,x4)=12μfx32+12μcx42+Ω2Mμf(μfx3+μcx4)2,
where the first and second terms of ζT are, respectively, the shifted-ectropy of the fuel and the coolant, and the third term is the shifted-ectropy of the total thermal-hydraulic loop. Differentiate ζT along the trajectory given by (25), and we can derive that
(41)ζ˙T=-Ω(x3-x4)2-2Mx42+P0x1x3+x1(μfx3θ3+μcx4θ4)-ΩMμf×{[2Mμfx3x4-2Mμcx42+P0x1(μfx3+μcx4)]gg+x1(μfx3+μcx4)(μfθ3+μcθ4)x42}=-Ωx32-Ω(1+2MΩ+2μcμf)x42+P0x1[(1+ΩM)x3+ΩμcMμfx4]+μfx1[(1+ΩM)x3+ΩμcMμfx4]θ3+μcx1[(1+ΩμcMμf)x3+ΩMx4]θ4.
Now, select the Lyapunov function for subsystem (25) as
(42)V1(x)=nr0ζT(x1,x2)+kf′(1+Ω/M)P0ζN(x3,x4),
and we can easily see that V1 is positive definite if and only if kf′ is a positive scalar. Then, from (38) and (41), the differentiation of V1 along the trajectory given by (25) and (28) can be represented as
(43)V˙1(x)=nr0ζ˙T(x1,x2)+kf′(1+Ω/M)P0ζ˙N(x3,x4)=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]+kf′x1[x3+Ωμc(M+Ω)μfx4]-x1(kf′x3+kc′x4)-x1ωTΓθ^+x1(Λθ1+βλx2θ2)+kf′x1(1+Ω/M)P0×{μf[(1+ΩM)x3+ΩμcMμfx4]θ3dd+μc[ΩMx3+(1+ΩμcMμf)x4]θ4}.
From (31), (32), and (33), we have
(44)V˙1(x)=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0×[x32+(1+2MΩ+2μcμf)x42]+x1θ~TΓω,
where
(45)θ~=θ-θ^
is the estimation error of uncertainty vector θ.
Choose the Lyapunov function for the closed-loop system formed by subsystem (25), virtual controller (28), and adaptive law (30) as
(46)V2(x,θ~)=V1(x)+12θ~TQΓθ~=nr0ζT(x1,x2)+kf′(1+Ω/M)P0ζN(x3,x4)+12θ~TQΓθ~.
Under assumption (26), we can obtain that
(47)θ~˙=θ˙-θ^˙=-θ^˙=-x1Q-1ω.
Differentiate V2 along the trajectory given by (25), (28), and (30), and then we have
(48)V˙2(x,θ~)=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42],
from which we can see that adaptive virtual control (28) provides globally asymptotic closed-loop stability for subsystem (25), if kf′ is positive. This completes the proof of this theorem.Remark 3.
From adaptive law (29), we have
(49)θ^=Q-1∫t0tx1ωdτ,
where t and t0 are, respectively, the current and initial time. Substitute (49) to control law (28), and it is clear that
(50)ξr=-knx1-kfx3-kcx4-ωTΓQ-1∫t0tx1ωdτ.
Based upon (50), this nonlinear adaptive controller takes the form as a PI control, and the integration term here is just introduced to compensate the uncertainty.Remark 4.
From (46), positive scalars qi(i=1,…,4) is the weighting factors of θ~i in V2. Here, qi is larger and the influence of V1 to V2 is smaller, which may lead to worse control performance.Nonlinear adaptive stabilizer (28) determines the referenced value of ξ, that is, the reactivity that should be induced by the control rods. In the following part of this subsection, this stabilizer is used to design control input u for entire system (6). Design and performance analysis of control input u is summarized as Theorem 8, which is the second main result in this paper.Theorem 5.
Consider control input of entire system (6)
(51)u=-kξ1x1-kξ2eξ+ξ˙r,
where both scalars kξ1 and kξ2 are positive, ξr is just the virtual control given in Theorem 2, and
(52)eξ=ξ-ξr.
Then, control input (51) can guarantee the globally asymptotic closed-loop stability.Proof.
Choose the Lyapunov function for the closed-loop system formed by system (6) and control input (51) as
(53)V3(x,θ~,eξ)=V2(x,θ~)+12kξ1eξ2=nr0ζT(x1,x2)+kf′(1+Ω/M)P0ζN(x3,x4)+12θ~TQΓθ~+12kξ1eξ2.
DifferentiateV3 along the trajectory given by system (6), control input (51), virtual control (28), and adaptive law (30), and we can derive that
(54)V˙3=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]+x1eξ+eξkξ1(u-ξ˙r)=-β(x1-x2)2(nr0+x1)(nr0+x2)-knx12-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]-kξ2kξ1eξ2.
From (54), state vector of entire system (6) converges to the origin asymptotically, which completes the proof of Theorem 5.


## 3.2. State-Observer Design

Although nonlinear adaptive control (51) guarantees the globally asymptotical convergence of the reactor state, the precondition is that state-vector x can be measured directly. Actually, x2 and x3, that is, variations of the relative concentration of delayed neutron precursor and the average fuel temperature, cannot be measured directly. Therefore, it is necessary to design a state-observer. Theorem 6 gives a bounded state-observer of system (6), which is the third main result.Theorem 6.
Suppose (26) is well satisfied, and moreover assume that
(55)cr=cr0+x2=nr0+x2≤C.
Consider the state-observer of system (6) that takes the form as
(56)x^˙=f(x^)+g(x^)ξ^+Koeo+Hoϑo,ξ^˙=u+koξ(nr0+x1)e1,
where x^=[x^1x^2x^3x^4]T∈R4 is the state-observation, ξ^ is the observation of ξ, and vector-valued functions f and g are determined by (11) and (12), respectively,(57)eo=[e1e4]T=[x1-x^1x4-x^4]T,(58)ϑo=[ϑ1ϑ4]T,(59)Ko=[kon+αfx^3+αcx^4+ξ^Λ2λ-P0μf[kof(nr0+x1)-1]-αcP0αfμfkof(nr0+x1)002Ωμfkoc]T,(60)Ho=[nr0+x1000000x1]T,(61)ϑ˙o=φ(x1,eo),(62)φ(eo,x1)=[(nr0+x1)e1x1e4]T,and positive scalars kon, kof, and koc are all observer gains. Observer (56) provides state-observation with bounded observing error that can be satisfactory small if both observer gains koN and koT are high enough.Proof.
Define the observer error vector as(63)e=[e1e2e3e4e5]=[xT-x^Tξ-ξ^]T.
From (6) and (56), it is clear that the observation error e satisfies
(64)Λe˙1=-β(e1-e2)+(nr0+x1)(αfe3+αce4+e5)+Λ(nr0+x1)ϑ~1-Λkone1,e˙2=λ(e1-e2)-2λe1+x1(nr0+x2)θ2,μfe˙3=P0e1-Ω(e3-e4)+P0[kof(nr0+x1)-1]e1-2Ωe4+μfx1θ3,μce˙4=Ω(e3-e4)-2Me4+μcx1ϑ~4-μckoce4+αcαfP0kof(nr0+x1)e1,e˙5=-koξ(nr0+x1)e1,
where
(65)ϑ~k=θk-ϑk,k=1,4.
Choose the Lyapunov function for observer error dynamics (64) as
(66)Ve1(e)=Λ2e12+β2λe22+|αf|2P0kof(μfe32+μce42)+12koξe52,
and by differentiating Ve1 along the trajectory given by (64), we can derive that
(67)V˙e1=-β(e12+e22)-Λkone12+(nr0+x1)(αfe1e3+αce1e4+e1e5)+Λ(nr0+x1)e1ϑ~1+βλ(nr0+x2)x1e2θ2+|αf|P0kof×[(e3+αcαfe4)-Ω(e32+e42)-2Me42-μde42+kofP0(nr0+x1)e1×(e3+αcαfe4)+μfx1e3θ3+μcx1e4ϑ~4]-(nr0+x1)e1e5=-β(e12+e22)-|αf|ΩP0kof[e32+(1+2MΩ)e42]-Λkone12-|αf|μckocP0kofe42+Λ(nr0+x1)e1ϑ~1+βλ(nr0+x2)x1e2θ2+|αf|x1P0kof(μfe3θ3+μce4ϑ~4).
Then, define the Lyapunov function of both observer (56) and parameter adaptive law (61) as
(68)Ve(e,ϑ~o)=Ve1(e)+12ϑ~oTΠϑ~o,
where
(69)ϑ~o=[θ1-ϑ1θ4-ϑ4]T,
and Π is a diagonal positive-definite matrix given by
(70)Π=diag([Λ|αc|μcP0kof]).
Moreover, since we have assumed that (26), it is clear that
(71)ϑ~˙o=-ϑ˙o=-φ(x1,eo).
Differentiate Ve along the trajectory given by observer (56) and adaptive law (71),
(72)V˙e=V˙e1+ϑ~oTΠϑ~˙o=-β(e12+e22)-|αf|ΩP0kof[e32+(1+2MΩ)e42]-Λkone12-|αf|μckocP0kofe42+Λ(nr0+x1)e1ϑ~1+βλ(nr0+x2)x1e2θ2+|αf|x1P0kof(μfe3θ3+μce4ϑ~4)-ϑ~oTΠφ(eo,x1)=-β(e12+e22)-|αf|ΩP0kof[e322+(1+2MΩ)e42]-Λkone12-|αf|μckocP0kofe42+βλ(nr0+x2)x1e2θ2+|αf|μfP0kofx1e3θ3≤-β(e12+e222)-|αf|ΩP0kof[e322+(1+2MΩ)e42]-β2[e2-1λ(nr0+x1)x1θ2]2-|αf|Ω2P0kof(e3-μfΩx1θ3)2+β2λ2(nr0+x2)2x12θ22+|αf|μf22ΩP0kofx12θ32-(Λkone12+|αf|μckocP0kofe42).
Based on both assumptions (26) and (55), it is clear from (72) that
(73)V˙e≤-eTMe+CMx12-(Λkone12+|αf|μckocP0kofe42),
where
(74)M=diag([ββ2|αf|Ω2P0kof|αf|ΩP0kof(1+2MΩ)0]),CM=β2λ2C2M22+|αf|μf22ΩP0kofM32.
From inequality (73), the observation error e enters to the ball given by
(75)Be={e∣eTMe≤CMx12-(Λkone12+|αf|μckocP0kofe42)}
from which we can see that if both kon and koc are larger, then the observation error is nearer to the origin. This completes the proof of Theorem 6.Remark 7.
From inequality (73), if
(76)limt→∞x1(t)=0,
then the observation error vector enters to the set
(77)Se={e∣e1=e2=e3=e4≡0}.
Further, from the last equation of observation error dynamics (64), e5 is a bounded constant.


## 3.3. Stability of Closed-Loop Stability

From Theorems2, 5, and 6, the nonlinear adaptive dynamic output-feedback power-level control strategy for the nuclear heating reactor can be written as
(78)u=-kξ1x1-kξ2eξ+ξ^˙r,ξ^r=-knx1-kfx^3-kcx4-θ^TΓω^,θ^˙=x1Q-1ω^,x^˙=f(x^)+g(x^)ξ^+Koeo+Hoϑo,ξ^˙=u+koξ(nr0+x1)e1,ϑ˙o=φ(x1,eo),
where
(79)ω^=[1x^2(1+ΩM)x^3+ΩμcMμfx4ΩMx^3+(1+ΩM)x4]T,
scalar kc is given by (31), matrices Γ, Q, Ko, and Ho are, respectively, given by (33), (35), (59), and (60), and vector-valued functions f, g, and φ are determined by (11), (12), and (62), respectively.In this section, we will study whether nonlinear adaptive control law (79) can provide the globally asymptotic stability of the reactor. The result for stability analysis of the closed-loop system formed by system (6) and control strategy (78) is summarized as Theorem 8 which is the fourth main result of this paper.Theorem 8.
Suppose that both (26) and (55) are well satisfied. Consider nonlinear adaptive dynamic output-feedback power-level control (78). If
(80)k-n=kn-β2λ2(1+C)2M22-kf2kofP02|αf|Ω×{1+μfP0(1-αfkf)[1+kof-1(1-kfαf)-1+Ωμf(M+Ω)μc(1-kfαf)-1](1-kfαf)-1}2M32
is a positive scalar, then the closed-loop system constituted by system (6) and control strategy (78) is globally asymptotically stable.Proof.
Similar to Theorems2, 5, and 6, Theorem 8 will also be proved by Lyapunov direct method. Select the Lyapunov function for the entire closed-loop system formed by (6) and (78) as
(81)V4(x,θ~,eξ,e,ϑ~o)=V3(x,θ~,eξ)+Ve(e,ϑ~o).
Differentiate V4 along the trajectory determined by both system model (6) and control (78),
(82)V˙4=V˙3(x,θ~,eξ)+V˙e(e,ϑ~o)=-knx12-β(x1-x2)2(nr0+x1)(nr0+x2)-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]+θ~TΓ(θ~˙+x1ω^)+βλθ2e2x1+[kf+kf′μf(1+Ω/M)P0(1+ΩM+ΩμcMμf)θ3]e3x1-kξ2kξ1eξ2-(β+Λkon)e12-βe22-|αf|ΩkofP0[e32+(1+2M+μckocΩ)e42]+βλ(nr0+x2)θ2e2x1+|αf|μfkofP0θ3e3x1=-β(x1-x2)2(nr0+x1)(nr0+x2)-kf′Ω(1+Ω/M)P0×[x32+(1+2MΩ+2μcμf)x42]-kξ2kξ1eξ2-(β+Λkon)e12-β2e22-|αf|ΩkofP0[e322+(1+2M+μckocΩ)e42]-β2(e2-1+nr0+x2λθ2x1)2-|αf|Ω2kofP0{e3-kof[kfP0|αf|Ω+μfΩ(1-kfαf)ddddddffffff×(1+kof-1(1-kfαf)-1ddddddffffffggghhh+Ωμc(M+Ω)μf)]gggggg×θ3x1(1-kfαf)}2-{kn-(1+nr0+x2)22λ2θ22-kofP02|αf|Ωgggig×[kf+kf′μfP0(1+kof-1(1-kfαf)-1dddkkkkkkkkkkk+Ωμc(M+Ω)μf)]2×θ32}x12≤-k-nx12-β(x1-x2)2(nr0+x1)(nr0+x2)-kf′Ω(1+Ω/M)P0[x32+(1+2MΩ+2μcμf)x42]-kξ2kξ1eξ2-(β+Λkon)e12-β2e22-|αf|ΩkofP0[e322+(1+2M+μckocΩ)e42],
where kf′ is defined in (34).
It is clear from inequality (82) that if k-n is a positive scalar, then the state of the closed-loop system finally enters into the set
(83)Θ={x∈R4,e∈R5,eξ∈R∣llllllllxk=0,ek=0,eξ=0,k=1,…,4},
which means that the corresponding closed-loop system is globally asymptotically stable. This completes the proof of this theorem.


## 4. Simulation Results with Discussion

To verify the feasibility and the performance of nonlinear adaptive dynamic output-feedback power-level control strategy (78), it is applied to the power-level regulation of an NHR with the thermal power of 200 MW. Both the closed-loop stability and load following ability are illustrated by numerical simulation results.


### 4.1. Description of the Numerical Simulation

The block diagram representation of the closed-loop constructed by the NHR model and the newly built nonlinear adaptive power-level control law is given in Figure1, where nr,m and nr,r are the relative neutron density given by the measurement and load signal, respectively, and Tcav,m and Tcav,r are the average coolant temperature determined by measurement and load signal, respectively. Here, the simulation model of the NHR, which was developed by INET, contains point kinetics model with six delayed neutron groups (7th order), 2nd-order heat exchange dynamics of the reactor core, 2nd-order dynamics of primary heat exchanger, 5th-order dynamics of a U-tube steam generator (UTSG) with a water-level controller and bypass system, 2nd-order dynamics of the pump of UTSG, and 6th-order dynamics of other heat transmission pipe or volume cells [30]. The water-level control of the UTSG adopts that one presented in [31].


### 4.2. Simulation Results

In this simulation, chooseQ=qI where q is a given positive scalar. Set kn=10.0, kf=0.1, kξ1=kξ2=0.1, kon=1000, koc=100, kof=0.01, and koξ=0.5. Here, the maximal control rod speed is 0.5 cm/s.Two case studies are done to show the control performance.Case A (large load lift).
Load signal changes linearly from 20% to 100% in 60 s:(1)
load lift with varyingkf and constant q;(2)
load lift with constantkf and varying q.Case B (load rejection).
Load signal steps down from 100% to 20% immediately:(1)
load lift with varyingkf and constant q;(2)
load lift with constantkf and varying q.


#### 4.2.1. Large Load Lift

This verification represents a hard operation for the NHR. In this case, the power demand changes linearly from 20% to 100% in 60 seconds. The responses of the relative nuclear power, average fuel temperature, and outlet coolant temperature of the reactor coreTcout and designed control rod speed with varying kf and constant q=500 are all illustrated in Figure 3. The responses of these process variables corresponding to varying q and constant kf=0.005 are all given in Figure 4.


#### 4.2.2. Load Rejection

This case also represents a stressed operation for the NHR. The load signal steps down from 100% to 20%. The responses of the process variables with varyingkf and q=500 are shown in Figure 5. The results corresponding to varying q and kf=0.005 are all given in Figure 6.


### 4.3. Discussion

It is clear that the load increase leadsδnr to be negative and decreasing and drives the power-level controller to give a positive control rod speed signal. Withdrawing of the control rod leads to both increase of nuclear power and that of fuel temperature. The closed-loop system enters into a steady state if the reactivity given by the control rods cancels that caused by the temperature feedback effect. As we can see from Figure 2, generation of the control rod speed signal is driven by the variations of relative nuclear power and average coolant temperature obtained from measurement. These two variation signals drive the state-observer to provide a convergent observation which then cause the power-level controller to generate the control rod speed signal.Figure 2
Block diagram representation of the closed-loop structure.Simulation results in Case A(1).
(a)
(b)
(c)
(d)Simulation results in Case A(2).
(a)
(b)
(c)
(d)Simulation results in Case B(1).
(a)
(b)
(c)
(d)Simulation results in Case B(2).
(a)
(b)
(c)
(d)From Figure3, if kf is larger, both transition time and overshoot of the fuel and outlet coolant temperatures are smaller. Actually, from (42) and (34), kf is larger, the weight of shifted-ectropy ζT in Lyapunov function V1 is larger, which induces higher response quality of the fuel and coolant temperatures. From Figure 4, if q is larger, then the dynamic performance of the key process variables deteriorates. The reason is given as follows. Based on (46), it is so easy to see that if q is larger, then the weight of V1 in Lyapunov function V2 is smaller. This induces a worse dynamic performance with a larger q. There even exist static control errors of both the fuel and coolant temperatures when q is large enough. Therefore, in the practical engineering, we should choose a larger kf and a smaller q to guarantee the load following performance.In case of power drop, the stepdown of the load signal causes the rapid increase ofδnr which results in the generation of a negative rod speed signal. The insertion of control rods then causes the decreases of both the nuclear power. Similarly in the case of load lift, the total closed-loop system enters into a steady state if the reactivity given by the control rods cancels that induced by the temperature feedback effect.From Figure5, it is easy to see that kf is larger, the transition time of the nuclear power, average fuel temperature, and the outlet coolant temperature are smaller. The reason is similar to that in the case of load lift. Actually, a larger kf leads to a higher weight of shifted-ectropy ζT in Lyapunov function V1, which induces the temperatures of the fuel and the coolant converges more quickly. Faster convergence in both fuel and coolant temperatures weakens their influence on the dynamic response of nuclear power through reactivity feedback effect, and we can see from Figure 5(a) that the transition time of the nuclear power is smaller and kf is larger. Moreover, if kf is enough high, we can see that there are larger overshoots of both the fuel and coolant temperatures, this is caused by the stepdown of the load signal which induce the rapid variations in δTcav. Then, feeding back this signal through a higher gain leads to a larger overshoot. Though the overshoots are larger when kf is larger, it is still acceptable.From Figure6, it is clear that q is larger, and the dynamic performance of the key process variables is worse. There are even static control errors of the fuel and coolant temperatures if q is large enough. Similar to the case of load lift, if q is larger, then the weight of V1 in V2 is smaller, which induces a worse dynamic performance.Finally, from the above analysis and discussion, we can see that nonlinear dynamic output- feedback power-level control law (78) guarantees globally asymptotic closed-loop stability and provides satisfactory load following performance by properly choosing the controller parameters. Due to the integration term induced by adaption mechanism, this nonlinear adaptive controller provides a high regulation performance. Based on (78), we can see that mathematic operations in this new control strategy are also addition, minus, proportional multiplication, differentiation, and integration, which is no more than those of the classical PID control. This also leads to the easy implementation of this control law in digital control system platforms. Moreover, it is worth to be noted that the power-level control strategy given in this paper is only for normal power operation of the NHR and NOT for reactor startup.


## 4.1. Description of the Numerical Simulation

The block diagram representation of the closed-loop constructed by the NHR model and the newly built nonlinear adaptive power-level control law is given in Figure1, where nr,m and nr,r are the relative neutron density given by the measurement and load signal, respectively, and Tcav,m and Tcav,r are the average coolant temperature determined by measurement and load signal, respectively. Here, the simulation model of the NHR, which was developed by INET, contains point kinetics model with six delayed neutron groups (7th order), 2nd-order heat exchange dynamics of the reactor core, 2nd-order dynamics of primary heat exchanger, 5th-order dynamics of a U-tube steam generator (UTSG) with a water-level controller and bypass system, 2nd-order dynamics of the pump of UTSG, and 6th-order dynamics of other heat transmission pipe or volume cells [30]. The water-level control of the UTSG adopts that one presented in [31].


## 4.2. Simulation Results

In this simulation, chooseQ=qI where q is a given positive scalar. Set kn=10.0, kf=0.1, kξ1=kξ2=0.1, kon=1000, koc=100, kof=0.01, and koξ=0.5. Here, the maximal control rod speed is 0.5 cm/s.Two case studies are done to show the control performance.Case A (large load lift).
Load signal changes linearly from 20% to 100% in 60 s:(1)
load lift with varyingkf and constant q;(2)
load lift with constantkf and varying q.Case B (load rejection).
Load signal steps down from 100% to 20% immediately:(1)
load lift with varyingkf and constant q;(2)
load lift with constantkf and varying q.


### 4.2.1. Large Load Lift

This verification represents a hard operation for the NHR. In this case, the power demand changes linearly from 20% to 100% in 60 seconds. The responses of the relative nuclear power, average fuel temperature, and outlet coolant temperature of the reactor coreTcout and designed control rod speed with varying kf and constant q=500 are all illustrated in Figure 3. The responses of these process variables corresponding to varying q and constant kf=0.005 are all given in Figure 4.


### 4.2.2. Load Rejection

This case also represents a stressed operation for the NHR. The load signal steps down from 100% to 20%. The responses of the process variables with varyingkf and q=500 are shown in Figure 5. The results corresponding to varying q and kf=0.005 are all given in Figure 6.


## 4.2.1. Large Load Lift

This verification represents a hard operation for the NHR. In this case, the power demand changes linearly from 20% to 100% in 60 seconds. The responses of the relative nuclear power, average fuel temperature, and outlet coolant temperature of the reactor coreTcout and designed control rod speed with varying kf and constant q=500 are all illustrated in Figure 3. The responses of these process variables corresponding to varying q and constant kf=0.005 are all given in Figure 4.


## 4.2.2. Load Rejection

This case also represents a stressed operation for the NHR. The load signal steps down from 100% to 20%. The responses of the process variables with varyingkf and q=500 are shown in Figure 5. The results corresponding to varying q and kf=0.005 are all given in Figure 6.


## 4.3. Discussion

It is clear that the load increase leadsδnr to be negative and decreasing and drives the power-level controller to give a positive control rod speed signal. Withdrawing of the control rod leads to both increase of nuclear power and that of fuel temperature. The closed-loop system enters into a steady state if the reactivity given by the control rods cancels that caused by the temperature feedback effect. As we can see from Figure 2, generation of the control rod speed signal is driven by the variations of relative nuclear power and average coolant temperature obtained from measurement. These two variation signals drive the state-observer to provide a convergent observation which then cause the power-level controller to generate the control rod speed signal.Figure 2
Block diagram representation of the closed-loop structure.Simulation results in Case A(1).
(a)
(b)
(c)
(d)Simulation results in Case A(2).
(a)
(b)
(c)
(d)Simulation results in Case B(1).
(a)
(b)
(c)
(d)Simulation results in Case B(2).
(a)
(b)
(c)
(d)From Figure3, if kf is larger, both transition time and overshoot of the fuel and outlet coolant temperatures are smaller. Actually, from (42) and (34), kf is larger, the weight of shifted-ectropy ζT in Lyapunov function V1 is larger, which induces higher response quality of the fuel and coolant temperatures. From Figure 4, if q is larger, then the dynamic performance of the key process variables deteriorates. The reason is given as follows. Based on (46), it is so easy to see that if q is larger, then the weight of V1 in Lyapunov function V2 is smaller. This induces a worse dynamic performance with a larger q. There even exist static control errors of both the fuel and coolant temperatures when q is large enough. Therefore, in the practical engineering, we should choose a larger kf and a smaller q to guarantee the load following performance.In case of power drop, the stepdown of the load signal causes the rapid increase ofδnr which results in the generation of a negative rod speed signal. The insertion of control rods then causes the decreases of both the nuclear power. Similarly in the case of load lift, the total closed-loop system enters into a steady state if the reactivity given by the control rods cancels that induced by the temperature feedback effect.From Figure5, it is easy to see that kf is larger, the transition time of the nuclear power, average fuel temperature, and the outlet coolant temperature are smaller. The reason is similar to that in the case of load lift. Actually, a larger kf leads to a higher weight of shifted-ectropy ζT in Lyapunov function V1, which induces the temperatures of the fuel and the coolant converges more quickly. Faster convergence in both fuel and coolant temperatures weakens their influence on the dynamic response of nuclear power through reactivity feedback effect, and we can see from Figure 5(a) that the transition time of the nuclear power is smaller and kf is larger. Moreover, if kf is enough high, we can see that there are larger overshoots of both the fuel and coolant temperatures, this is caused by the stepdown of the load signal which induce the rapid variations in δTcav. Then, feeding back this signal through a higher gain leads to a larger overshoot. Though the overshoots are larger when kf is larger, it is still acceptable.From Figure6, it is clear that q is larger, and the dynamic performance of the key process variables is worse. There are even static control errors of the fuel and coolant temperatures if q is large enough. Similar to the case of load lift, if q is larger, then the weight of V1 in V2 is smaller, which induces a worse dynamic performance.Finally, from the above analysis and discussion, we can see that nonlinear dynamic output- feedback power-level control law (78) guarantees globally asymptotic closed-loop stability and provides satisfactory load following performance by properly choosing the controller parameters. Due to the integration term induced by adaption mechanism, this nonlinear adaptive controller provides a high regulation performance. Based on (78), we can see that mathematic operations in this new control strategy are also addition, minus, proportional multiplication, differentiation, and integration, which is no more than those of the classical PID control. This also leads to the easy implementation of this control law in digital control system platforms. Moreover, it is worth to be noted that the power-level control strategy given in this paper is only for normal power operation of the NHR and NOT for reactor startup.


## 5. Conclusion

Nuclear heating reactor (NHR) is a small reactor with many advanced safety features and has a promising future in the fields of district heating, seawater desalination, and electricity production. It is so clear that the NHR dynamics has the features of high nonlinearity and uncertainty, which leads to the need of developing nonlinear adaptive power-level control. In this paper, a novel nonlinear adaptive power-level control strategy is established based upon the physically based control design approach. It has already been proved theoretically that this newly build controller guarantees globally asymptotic closed-loop stability and satisfactory load following performance. This new control law is then applied to the power-level control of a nuclear heating reactor. Numerical simulation results show not only the feasibility of this controller but also the relationship between the performance and the tuning parameters. This new nonlinear adaptive dynamic output-feedback power-level regulator can be easily implemented on those digital control system platforms. Finally, since the NHR is essentially a kind of PWR, the results in this paper can be directly applied to the power-level control of PWRs. The future work is to design a more powerful adaptive dynamic output-feedback power-level control law based on artificial neural networks [32] or fuzzy sets [33].



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*Source: 794167-2013-08-05.xml*