Over the past century, human beings have actively carried out various space explorations. Remarkable accomplishments have been achieved, which continuously expanded the boundaries of human knowledge. Many of these accomplishments going to the unknown outer and deep space are largely benefited from the successful development and application of solar cell battery or space radio-isotopic battery, which exploits the energy either from the solar irradiation (Baraskar et al., 2022) or the energy released during decay process of certain radioactive isotopes (Bennett et al., 1996) to provide the necessary power supply. In addition to continuing the deep-space exploration, the next important step will be surface explorations of planetary satellites and other-than-earth planets, among which lunar exploration projects have already been carried out by many countries or international organizations, see for instance in the reviews by Zheng et al. (2008), Ehrenfreund et al. (2012) and Marov and Slyuta (2021). Manned surface exploration of the Mars is also undergoing the process of intense discussions (McNutt et al., 2015). For surface exploration of other-than-earth planets or planetary satellites, traditional solutions of power supply based on solar cell battery or radio-isotopic battery will not be able to meet the needs due to the impact of weak solar light, extreme high or low temperature, dust storms, as well as the demand of a power supply in hundreds of kilowatts or even megawatts and a mission task cycle that can easily last more than 10 years or even decades (Akimov et al., 2012).

Thanks to its unique advantages of high power-to-mass ratio, shallow reactivity poisoning and quick response to reactivity control (IAEA, 2002), a liquid metal-cooled space nuclear reactor, which adopts a fast neutron spectrum reactor core cooled by liquid metal or liquid metal alloys, is the preferred solution to provide the necessary high-power supply also in the environment of lacking or no solar irradiation. As recently reviewed in (Song et al., 2021), worldwide 80% of the prototype designs and 100% of the actual tested space power supply systems are based on various designs of liquid metal-cooled space nuclear reactors. Thanks to its high boiling point and low density, liquid lithium has attracted more and more attentions as the coolant for space nuclear reactor from both industrial and scientific community, see for instance in (Harty and Mason, 1993; Demuth, 2003; Jin, et al., 2022) etc. However, to meet the requirements of safety, reliability, viability, and the long-life expectancy of a space exploration mission, autonomous intelligent control of the reactor are of vital importance when designing the control system of a space nuclear reactor (Zhao at al., 2012). In terrestrial nuclear power plants (NPPs), human operators can manually perform the necessary control functions required for normal and abnormal operating conditions. On the contrary, for space missions into outer and deep space characterized by uncertain environments, rare events, and communication delays, all control functions of the reactor must be performed through a robust control system with very limited or even no human intervention from the Earth (Zhao at al., 2012). The basic requirement is to achieve a fast and accurate control of the reactor core power following the desired power output in different operation modes, including not only steady-state operation at different power levels, but more importantly, also transients between these power levels, and transient modes during startup and emergency shutdown of the reactor.

Thanks to its advantages of stability and maturity, the classical proportional-integral-derivative (PID) control scheme is still widely installed as the standard core-power control scheme in current PWR power plants. However, PID control scheme is not able to well handle the load-following operation modes of a nuclear reactor (Liu et al., 2009). In the past decades, various techniques to enhance the control performance of nuclear reactor have been extensively studied. Advanced control schemes including fuzzy control (Mamdani, 1974), neural network control (Mamdani, 1974), fuzzy PID control (Liu et al., 2009; Zeng et al., 2021) etc. have been proposed. But it is very difficult to design an optimized controller for a nuclear reactor due to its high system complexity and nonlinearity, continuous variations of the reactor parameters with operation time and burn up levels, as well as the inevitable modeling uncertainties of real plant behaviors. In this regard, the methodology of model predictive control (MPC), which has received increased attention as a powerful tool for the control of industrial processing systems (Richalet et al., 1978; Garcia et al., 1989), was also proposed as a suitable strategy for the control of nonlinear, time-varying systems such as nuclear reactors thanks to its unique characteristic of online rolling optimization (Na, 2001; Na et al., 2003; Na et al., 2005; Liu and Wang, 2014). The basic concept of MPC is the utilization of a predictive model describing input, output, and state constraints of the objective system, to solve, at each sampling timestep, an online optimization problem to determine the optimal control actions for a finite future counting from the current time instant (Liu and Wang, 2014). However, once a future trajectory of control moves has been obtained, only the first element of that trajectory is employed as input of the controller. The same optimization calculation will be then conducted at each subsequent time instants. This method has many advantages over the conventional infinite horizon optimal control because it is a suitable control strategy for nonlinear, time-varying systems (Na et al., 2006; Eliasi et al., 2012).

It should be noted, for nonlinear, time-varying systems such as a nuclear reactor core, model predictive controller can be readily defined based on state-space model linearized at locally defined steady-state working points. However, a non-distortive description of the system dynamics can be guaranteed only in the vicinity of these points. Hence, the local predictive controller is only meaningful to system dynamic process near these specific points. Consequently, severe mismatch between the model and the actual plant will significantly worsen the control effect when the control range expands beyond these specific steady-state working points. Fortunately, the nonlinear model of a nuclear reactor core can be divided into multiple linear models according to different core power levels. The reactor power control is a typical so-called multi-model control problem, for which a fuzzy “integration” based on the T-S fuzzy theory (Takagi and Sugeno, 1985) could be employed. Nonlinear feature of the global system can be approximated by a fuzzy “integration” using multiple linear features carefully designed at specific local steady-state working points according to the so-called parallel distributed compensation (PDC) scheme. This methodology of combining model predictive control and fuzzy integration is termed in this study as the so-called fuzzy model predictive control (FMPC). The basic idea is that for each locally linearized state-space model, a MPC controller is designed. Afterwards, the overall controller, which is by its nature still nonlinear, is then obtained via a fuzzy “blending” of all the linear controllers individually designed at local steady-state working points (Ma and Sun, 2000).

In this study, the concept of FMPC was adopted to design the core-power controller of a megawatt ultra-small lithium-cooled space nuclear reactor (Song et al., 2021), which is a prototype design for the space exploration program developed in the framework of the National Key R&D Program of China. Full rated power (FP) of the liquid lithium cooled reactor core is 5.1   M W . The nonlinear reactor core model consisting of equations describing point neutronic kinetics, thermohydraulic behaviors as well as reactivity feedbacks were first linearized at five steady-state working points of relative core power of 20 % , 40 % , 60 % , 80 % and 100 % F P , respectively. For each of the five local state points, a local core-power MPC controller was established based on the linearized state-space equations as the predictive model. The overall core-power controller applicable to the full range of nonlinear reactor core, which is in its nature still nonlinear, was then constructed by a weighted combination of the five local MPC controller via fuzzy integration by using the membership functions specially defined for the selected five steady-state working points. To verify the performance of the FMPC core-power controller, effective, robust, and accurate control of the reactor core power was then realized for five predefined, representative operation transients to space nuclear reactor, including ramp and step transients, as well as disturbance transient. Simulation results show that the intelligent FMPC controller possesses an excellent load-following ability and anti-interference ability, both of which are of vital importance for space exploration missions. When compared with the classical PID controller, the FMPC controller designed in this study shows also a much better performance with smaller overshoot, lesser adjusting time and lower integral time-squared error (ITSE). This study successfully demonstrates the feasibility of FMPC to design sophisticate and effective core-power controller also for nuclear reactors with a fast neutron spectrum.

Fuzzy model predictive control (FMPC) is a modern intelligent control methodology combining model predictive control (MPC) (Garcia et al., 1989) and T-S fuzzy theory (Takagi and Sugeno, 1985). In this section, a brief description of model predictive control and T-S fuzzy theory will be provided.

In the framework of the National Key R&D Program of China, a prototype design of a megawatt ultra-small lithium-cooled space nuclear reactor to meet the requirements of inherent safety, light weight and long lifetime for typical outer and deep space exploration missions was presented (Song et al., 2021). Principle design of the reactor core is depicted in Figure 3, which adopts liquid lithium circulated by an electromagnetic primary pump as the coolant to transfer the heat released in the reactor core to the secondary Brayton cycle via the primary heat exchanger. In this study, the basic neutronic and thermohydraulic characteristics of the liquid lithium-cooled reactor core were adopted as the object of investigation, for which the most important neutronic and thermohydraulic parameters of the reactor core model are summarized in Table 1. A brief description is also given in the following:

(1) According to the design concept, full rated power (FP) of the reactor core P n 0 is 5.1 M W and the inlet coolant temperature of the core T c , i n 0 is 1,450 K .

Based on the above lumped neutronic and thermohydraulic parameters, the following mathematical model describing the most essential neutronic and thermohydraulic behavior of the liquid lithium-cooled reactor core was able to be established, which yields: { d n r d t = ρ − β Λ n r + β Λ c r d c r d t = λ n r − λ c r d T f d t = P n 0 μ f n r − Ω μ f T f + Ω 2 μ f T c , o u t + Ω 2 μ f T c , i n d T c , o u t d t = 2 Ω μ c T f − 2 M + Ω μ c T c , o u t + 2 M − Ω μ c T c , i n (14) In the above equation system,

(1) t is the time coordinate.

Due to the feedback reactivity by fuel and coolant temperature as given in Eq. 15, the reactor core model established in Eq. 14 describing the most essential neutronic and thermohydraulic behavior of the liquid lithium-cooled reactor core is, by its nature, a nonlinear equation system. Before designing a core-power controller according to the concept of model predictive control, the reactor core model needs to be first linearized, for which the following consideration were taken:

(1) At a given moment t 0 , the reactor core is assumed at a steady state characterized by n r = n r 0 , c r = c r 0 , T c , i n = T c , i n 0 , T c , o u t = T c , o u t 0 , T f = T f 0 .

With the above assumptions, the reactor core model given in Eq. 14 is linearized at the steady-state working point n r 0 , which yields: { d ∆ n r d t = − β Λ ∆ n r + β Λ ∆ c r + α f n r 0 Λ ∆ T f + α c n r 0 2 Λ ∆ T c , o u t + n r 0 Λ ∆ ρ r o d d ∆ c r d t = λ ∆ n r − λ ∆ c r d ∆ T f d t = P n 0 μ f ∆ n r − Ω μ f ∆ T f + Ω 2 μ f ∆ T c , o u t d ∆ T c , o u t d t = 2 Ω μ c ∆ T f − 2 M + Ω μ c ∆ T c , o u t (16)

Furthermore, the state-space variables x , control moves u and output y are then defined as the follows: x = [ ∆ n r , ∆ c r , ∆ T f , ∆ T c , o u t ] T u = [ ∆ ρ r o d ]   y = ∆ n r (17)

Finally, the linearized model is given as: { x ˙ = A c x + B c u y = C c x + D c u (18) where the coefficients A c , B c , C c and D c are given as the follows: A c = [ − β Λ β Λ n r 0 α f Λ n r 0 α c 2 Λ λ − λ 0 0 P n 0 μ f 0 − Ω μ f Ω 2 μ f 0 0 2 Ω μ c − ( 2 M + Ω ) 2 μ c ] (19) B c = [ n r 0 Λ ,   0 ,   0 ,   0 ] T C c = [ 1 ,   0 ,   0 ,   0 ] D c = [ 0 ]

Eq. 18 with the coefficients given in Eq. 19 is the form of the linearized reactor-core model at the steady-state working point characterized by n r 0 . To meet the requirements of model predictive control algorithm, Eq. 18 is then discretized, which yields: { x ( k + 1 ) = A x ( k ) + B u ( k ) y ( k ) = C x ( k ) (20)

The coefficients A , B and C are given as: A = e A c T B = B c ∫ 0 T e A c T d t C = [ 1 ,   0 ,   0 ,   0 ] (21) in which T is the sampling period. Eq. 20 is the discretized form of the linearized reactor core model at the steady-state working point characterized by n r 0 , which is then used as the predictive model corresponding to this steady state.

In this study, a fuzzy model predictive control based on model predictive control and T-S fuzzy theory was designed for the core-power control of a megawatt ultra-small lithium-cooled space nuclear reactor. Firstly, neutronic and thermohydraulic behaviors of the lithium-cooled reactor core described by nonlinear state-space equations were linearized at five steady-state working points characterized by relative core power of 20 % , 40 % , 60 % , 80 % and 100 % F P , respectively. Afterwards, a local model predictive controller was designed for each of the five local steady-state working points based on the state-space equations locally linearized at the specific state points. Finally, all the five local MPC controllers were integrated based on T-S fuzzy theory via the so-called parallel distributed compensation scheme, to build up the so-called fuzzy model predictive control for overall control of the nonlinear reactor core. Five typical operation conditions including both ramp and step transients, as well as a disturbance transient were defined and simulated to assess the performance of FMPC core-power controller. The simulation results show that the FMPC controller has an excellent load-following ability and anti-interference ability, both of which are of vital importance for space exploration missions. When compared with the classical PID controller, the FMPC controller designed in this study shows also a much better performance with smaller overshoot, lesser adjusting time and much lower ITSE values.

Although investigations in this study were conducted for a simplified neutronic and thermohydraulic model of a lithium-cooled space nuclear reactor, it successfully demonstrates the feasibility of fuzzy model predictive control to design sophisticate and effective core-power controller also for typical nonlinear and time-varying nuclear reactors with a fast neutron spectrum. The design method adopted in this study for the core-power controller can be readily extended to other designs of space nuclear reactors, provided more in-depth detailed specifications of specified designs are become available in the future. Nevertheless, it should be pointed out, physical constraints regarding driving mechanism of the control rods are not available for the prototype design of the lithium-cooled space nuclear reactor. Therefore, control constraint is not considered in the FMPC controller designed in this study, which, however, should be included in further investigations if the driving mechanism of the control rods become available. Furthermore, applicability of the fuzzy model predictive control is still worthy further investigations for a wide range of realistic working scenarios of space nuclear reactors, especially for those scenarios of large and abrupt disturbances to the reactor, provided they are become available in the future.