Recently, concerns have been growing about the intensification and increase in extreme weather events, including torrential rainfall and typhoons. For mitigating the damage caused by weather-induced disasters, recent studies have started developing weather control technologies to lead the weather to a desirable direction with feasible manipulations. This study proposes introducing the model predictive control (MPC), an advanced control method explored in control engineering, into the framework of the control simulation experiment (CSE). In contrast to previous CSE studies, the proposed method explicitly considers physical constraints, such as the maximum allowable manipulations, within the cost function of the MPC. As the first step toward applying the MPC to real weather control, this study performed a series of MPC experiments with the Lorenz 63 model. Our results showed that the Lorenz 63 system can be led to the positive regime with control inputs determined by the MPC. Furthermore, the MPC significantly reduced necessary forecast length compared to earlier CSE studies. It was beneficial to select a member that showed a larger regime shift for the initial state when dealing with uncertainty in initial states.

In recent years, concerns have been raised regarding the intensification and increase in extreme weather events such as torrential rainfall and typhoons. To mitigate the damage caused by weather-induced disasters, efforts have been made to improve the forecasting accuracy of stationary heavy rainfall and develop disaster prevention infrastructures, including dams and embankments. Recently, Japan's Moonshot program started exploring alternative countermeasures for mitigating weather-induced disasters. Specifically, the program aims at developing weather control technologies to lead the weather to a desirable regime with feasible manipulations. Under the program, researchers are exploring various engineering techniques such as cloud seeding and atmospheric heating. However, the possible magnitude of humans' manipulations of the atmosphere is limited. Therefore, simulation studies using numerical weather prediction (NWP) models are needed in addition to the engineering studies to develop effective control approaches with feasible manipulations.

To date, a few simulation studies with NWP models have been conducted for mitigating extreme events. For example, Henderson et al. (2005) conducted numerical experiments using a modified version of the Fifth-Generation Penn State/NCAR Mesoscale Model (MM5) 4D-Var to identify the temperature increments required for minimizing wind-related damage caused by Hurricane Andrew in 1992. However, the results may not be sufficiently realistic due to various experimental limitations (Henderson et al., 2005). The Typhoon Science and Technology Research Center of Yokohama National University proposed using sailing ships and artificial upwelling to reduce the intensity of tropical cyclones. Their simulations demonstrated that the drag enhancement caused by the sailing ships and sea surface temperature decrease by the artificial upwelling successfully weakened tropical cyclones (Hironori Fudeyasu; personal communication, 2023). Previous studies, however, examined impacts of the manipulations on specific extreme events through control experiments that simply compared simulations with and without manipulations. Here, a research framework is necessary to develop effective control approaches with feasible manipulations.

Miyoshi and Sun (2022; hereafter MS22) proposed a control simulation experiment (CSE), an experimental framework for systematically evaluating and exploring control approaches with unknown true values by expanding the observing systems simulation experiment (OSSE). They conducted CSEs with the three-variable Lorenz 63 model (Lorenz, 1963) and succeeded in leading the system to the positive regime with small control inputs. Sun et al. (2023; hereafter SMR23) also applied to CSEs for the 40-variable Lorenz 96 model (Lorenz, 1996), showing that their CSEs succeeded in reducing the number of extreme events of the Lorenz 96 model. Furthermore, Ouyang et al. (2023; hereafter OTK23) successfully reduced the total magnitude of control inputs with the Lorenz 63 model by approximately 20 % compared to MS22's approach by regulating the amplitude of control inputs based on the maximum growth rate of the singular vector. The previous CSE studies (MS22, SMR23, and OTK23) generated control inputs as differences between ensemble members that stay within and those that deviate from the desired regime. However, physical constraints, generally needed for real-world applications, cannot be explicitly considered in previous CSE studies. Therefore, it is worthwhile to explore other methodologies to determine control inputs.

In this study, we propose introducing the model predictive control (MPC) within the framework of CSE. The MPC is an advanced control method that repeats prediction and optimization with explicit consideration of constraints. While the MPC has been widely used in practical fields such as the process industry and power electronics (Schwenzer et al., 2021), there has been no study yet that used the MPC for mitigating weather-induced disasters to the best of our knowledge. As the first step toward applying the MPC to the real weather control, this study performs a series of MPC experiments with the Lorenz 63 model. Here we explore the way to implement the MPC within CSE and aim to reveal important issues to extend the MPC to high-dimensional NWP models.

The remaining sections of this paper are arranged as follows. Section 2 introduces the theory of the MPC and describes the experimental setting. In Sect. 3, we employ a series of MPC experiments with the Lorenz 63 model and discuss properties of the MPC applied to the chaotic dynamical systems. Finally, Sect. 4 provides a summary.

This study explores using the MPC for controlling a chaotic dynamical system. Here, the MPC is a feedback control method that identifies control inputs to minimize the cost function under constraints at each time. In other words, the MPC is a control method that solves an optimal control problem (OCP) for a finite horizon at each time. Strictly speaking, the MPC considered in this study is nonlinear model predictive control (Chen and Shaw, 1982; Keerthi and Gilbert, 1988; Mayne and Michalska, 1990; Mayne et al., 2000).

First, we define the terminology and symbols. As shown in Fig. 1, the two key processes of the MPC are model-based prediction and optimization of control inputs in the OCP. For these processes, the prediction horizon,

Next, we describe the procedure of the MPC. First, the MPC requires the suitable design of a numerical model,

The present state

The predicted state,

Based on

Prediction (step 2) and optimization (step 3) are iterated with updated

The control input,

The process returns to step 1 and repeats these steps at

Conceptual image of the model predictive control (MPC; the gray block). A numerical model,

As previously noted, the MPC identifies control inputs that allow the system to achieve a desirable state for a finite horizon by solving the OCP at each time. Here, we explain that the OCP can be regarded as a variational problem with constraints. We consider a basic OCP with control and prediction horizons being

The following necessary conditions for optimal control inputs are obtained by converting the constrained problem to an unconstrained problem using the method of Lagrange multipliers (see Appendix A):

When the control horizon is shorter than the prediction horizon (

This study uses the Lorenz 63 model for MPC experiments. The Lorenz 63 model is a three-variable nonlinear differential equation expressed as follows:

This study considers a control problem, namely keeping the Lorenz 63 system in the positive regime (

The CSE is an experimental framework that controls the nature run (NR), extended from OSSE. The key concept of CSE is that the true state of the NR is unknown, but manipulations can be added to the NR, assuming a realistic atmosphere.

Based on previous studies (Kalnay et al., 2007; Yang et al., 2012; MS22; OTK23), the experimental setting of our CSE is determined as follows. We first employ a free run with the Lorenz 63 model for 2 009 000 steps without any manipulations. The initial values of the free run are generated by random numbers

We employ three indicators to evaluate CSEs. The first index is the success rate (SR), which denotes the percentage of cases that satisfy

The procedure of the CSE with MPC is designed as follows:

At a certain time

DA is employed to obtain an analysis ensemble,

The ensemble forecast,

If at least one member indicates a regime shift (RS) during the ensemble forecast, the process continues to step 5. Otherwise, the NR evolves until

The OCP is solved to obtain control input

The NR is evolved from

The process returns to step 1 and repeats these steps at

This procedure is illustrated in Fig. 2. For simplicity, the flow diagrams of the CSE are divided into two cases: without a RS in Fig. 2a and with a RS in Fig. 2b. The procedure of the CSE for forecasts without a RS in Fig. 2a is identical to the OSSE. In contrast, the procedure of the CSE for forecasts with a RS in Fig. 2b has additional processes for identifying and applying control inputs. The upper illustration in Fig. 2c shows a conceptual image of identifying control inputs, and the lower illustration shows an application of control inputs to the NR through the Lorenz 63 model. Importantly, the NR cannot be used as the initial state of the OCP because it is always unknown. Therefore, an analysis ensemble is used as the initial state. As discussed later (Sect. 3.5), the initial state for the OCP substantially affects the control results, and the member with the smallest state

The RMSEs and the multiplicative inflation parameters used in this study for each ensemble size,

Flow diagram and conceptual image of the CSE with MPC.

First, CSE is conducted with the Lorenz 63 model to verify the impacts of the MPC on the NR. The control objective is leading the system to the positive regime under the minimization of the three-variable control inputs. Here,

The NR and the

The NR and controlled NR of the Lorenz 63 model for 2000 steps. Each starting point is selected from the 24th step of 2 000 000-step DA cycles. Panel

The controlled NR and the

Figure 5 shows the prediction of the state and optimization of the control inputs in each horizon at an arbitrary selected step (the 232nd step of the CSE of Fig. 4). Since the forecast (dashed blue line) from the initial state shows a RS, the control is activated to solve the OCP. As demonstrated in Fig. 5a, the trajectory of the controlled prediction gradually shifts to satisfy

The prediction of the state and optimization of the control inputs in the horizon at an arbitrary selected step (the 232nd step of the CSE of Fig. 4). Iterative computations were performed 356 times for solving the OCP in this case.

Here, we investigate the sensitivity to

It should be noted that a higher SR does not necessarily indicate lower MTF. For example, focusing on

Hereafter, the experiment with

Sensitivity to the prediction horizon,

Summary of the success rate (SR), mean total failure (MTF), and mean total control inputs (MTCIs) for results in each experimental setting, with

For realistic control scenarios, it is important to consider control problems in which limited control inputs relative to model dimensions are available. Here, this section investigates the CSE with one-variable control input.

Figure 7a, b, and c show the NRs controlled only by

The NRs controlled by one-variable control input:

Here, we show that the MPC can consider constraints for control inputs in addition to the constraint for state (i.e.,

Figure 8 shows the NRs and the

MPC experiments with inequality constraints for control inputs with

For controlling NRs, it would be preferable to use the NR as the initial state for identifying control inputs. However, the state estimated by DA must be used because the true value is always unknown. Therefore, there is uncertainty in MPC-derived control inputs based on the states estimated by DA. This uncertainty may not cause serious problems for some systems without strong nonlinearity. Chaotic dynamical systems, however, require careful explorations of options for stable control because small uncertainties can cause large differences. Here, we discuss the initial state that would be valid for leading a chaotic dynamical system to a prescribed regime.

We performed 1000 independent CSEs and computed the SR, MTF, and MTCIs for five kinds of initial states: random (all mem.), mean (all mem.), random (RS mem.), mean (RS mem.), and largest (RS mem.), respectively. The (all mem.) label denotes selection among all members in the analysis ensemble, and the (RS mem.) label denotes selection among the members of the analysis ensemble showing RSs. The random label denotes a randomly sampled member, the mean label denotes the mean of the members, and the largest label denotes the member showing the largest RS. For example, mean (all mem.) indicates the mean analysis ensemble. The results are shown in Fig. 9. The experiment with the largest (RS mem.) state yielded the best results, showing the highest SR and the smallest MTF and MTCIs. Furthermore, Fig. 9 shows that it is better to use a member selected from the (RS mem.) group rather than (all mem.) as the initial state. We presume that it is safer to select a member showing a larger RS for the initial state when uncertainty exists in initial state. Therefore, the improvement with a larger ensemble size,

Sensitivity to the initial state, with

In this study, we propose introducing the MPC within the framework of CSE. The advantage of using the MPC is that control objectives and constraints can be explicitly considered. Therefore, we expect that this approach will be useful for realistic weather control by designing a suitable cost function and constraints.

We conducted MPC experiments with the Lorenz 63 model and successfully led the system to the positive regime. The previous CSE studies (MS22 and OTK23) required longer forecasts (about 300 steps) for successful controls with the Lorenz 63 model, whereas our approach required much shorter forecasts, such as with 20 steps. We also confirmed that controllability would be difficult with limited variables of control inputs or with additional constraints. In our discussion, we suggest that it is safer to select a member showing a larger RS for the initial state when dealing with uncertainty in initial states.

This study is an investigation of the first phase of the MPC for weather control. In the future, this approach will be investigated with more realistic NWP models. In addition, several improvements remain for the MPC to be applied to weather control. Our present approach requires many iterations to solve the OCP, and temporal forward and backward computations are required for each iteration. This means that it is computationally difficult to apply the present approach to high-dimensional NWP models as it is. Therefore, further studies are needed to explore faster approaches to solve OCPs for high-dimensional models. For this challenge, we expect the continuation/generalized minimal residual (C/GMRES) method (Ohtsuka, 2004) and quantum annealing (Inoue and Yoshida, 2020) to be fast solvers for the MPC. Furthermore, we need to consider a variety of uncertainties such as model errors and weather shifts during identifying control inputs. Therefore, uncertainty quantification is also an important research topic prior to real-world field experiments.

Finally, we emphasize caution in weather control research. The achievement of control for extreme events would be an innovative way to mitigate weather-induced disasters. However, the side effects of weather control must be carefully examined from an ethical, legal, and social issues (ELSI) perspective. In particular, we need to discuss not only the destructive side effects caused by control failures, but also the impact on biodiversity and many industries (e.g., electricity production). Our research program also addresses such social issues with legal and ethical researchers. Further ELSI research will also be conducted to satisfy responsible and innovative research for weather control studies.

Here, we derive the necessary conditions for optimal control inputs. For simplicity, we consider the following problem:

The Lorenz 63 system is known to increase the amplitude of

Table B1 compares the SR, MTF, and MTCIs of 1000 independent CSEs for two starting point settings (i.e.,

Comparison of the success rate (SR), mean total failure (MTF), and mean total control inputs (MTCIs) for two starting point settings, with

The code that supports the findings of this study is available from the corresponding author upon reasonable request. In addition, all of the data and codes used in this study are stored for 5 years at Chiba University.

The authors declare that all data supporting the findings of this study are available within the figures and tables of the paper.

FK and SK conceptualized this study. FK conducted the numerical experiments and wrote the paper. SK supervised and directed this study.

The contact author has declared that neither of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

The authors thank members of the Moonshot R&D program for valuable discussions.

This study was partly supported by the Japan Science and Technology Agency Moonshot R&D program (grant nos. JPMJMS2284 and JPMJMS2389), the Japan Society for the Promotion of Science (JSPS) via KAKENHI (grant no. JP21H04571), and the IAAR Research Support Program of Chiba University.

This paper was edited by Pierre Tandeo and reviewed by two anonymous referees.