MPC is a control strategy that optimizes control inputs by using a dynamic model to predict the future behavior of the system. MPC solves an optimization problem at each time step to minimize a cost function over a finite predictive horizon. The specific design of the cost function depends on the application, but the general formulation can be expressed as follows: 1Ju0,u1,…,uTc=∑t=0Tcut⊤Cu-1ut︸Jinput+∑t=0Tprt-Hcxt⊤Cr-1rt-Hcxt︸Jstate.s.t.xt+1=Mtxt,ut. Here, xt denotes the state variable at time t. The next state xt+1 is obtained by integrating the nonlinear forecast model operator Mt forward from the current state xt and the control input ut. The control input cost Jinput is typically optimized over a shorter control horizon Tc within the prediction horizon Tp. Jinput penalizes the magnitude of the control input, preventing it from being excessively large. The state deviation cost Jstate evaluates the difference between model-predicted states and the control objective r, and the optimization problem is performed over a finite prediction horizon Tp. Hc is an operator that maps the state variables x to the control variables. Cu and Cr are weighting matrices for the control input u and the deviations between state variables and the control objective, respectively. In this study, the control horizon Tc is shorter than the prediction horizon Tp, where control is applied only at the first time step of each cycle.

In conventional MPC, optimal control inputs are typically obtained by minimizing a cost function through gradient-based optimization. For nonlinear systems, this often involves solving the adjoint equations to efficiently compute gradients of the cost function with respect to control variables. Although this approach is accurate, it requires derivation and implementation of the adjoint model, which can be costly and challenging, especially for high-dimensional systems such as numerical weather prediction models.

Among the two components of the cost function in Eq. (), the state deviation cost Jstate typically has the highest computational cost. This is because it involves predicting and evaluating the future states of the system over the entire prediction horizon, which requires extensive computations, especially for complex or nonlinear systems. The ensemble approximation can mitigate this computational cost by using representative trajectories to approximate future states, as discussed in Sect. .

The 4DVar method estimates the optimal initial state x0 over a time window by considering the misfits between observations and forecast model states at multiple times. This process is achieved by minimizing the following cost function : 2Jx0=x0-x0b⊤B-1x0-x0b︸Jbackground+∑t=0τyt-Hxt⊤R-1yt-Hxt︸Jobservation,s.t.xt+1=Mtxt. The first term in Eq. () qualifies the difference between the initial guess (background or prior) x0b and the estimated state x0, weighted by the background error covariance matrix B. The second term in Eq. () measures the misfit between the state variables and the observations y at times t=0,1,2,…,τ. The observation operator H maps the state x to the observation space, and R represents the observation error covariance matrix. The time window τ is referred to as the data assimilation window and plays the same role as the prediction horizon Tp in MPC. Therefore, the second term Jobservation in Eq. () serves a similar purpose to the state deviation cost Jstate in the MPC cost function (Eq. ) as both evaluate the discrepancies between the predicted states and the target values or observations over a specific time horizon.

Operational systems often implement 4DVar using an incremental approach to utilize the linearized model instead of the full nonlinear model . Defining δx0=x0-x0b, the cost function J(x0) in Eq. () becomes 3Jδx0=δx0⊤B-1δx0︸Jbackground+∑t=0τHδxt-dt⊤R-1Hδxt-dt︸Jobservation,s.t.δxt+1=Mtδxt, where Mt and H are the tangent linear operators of Mt and H, respectively. The innovation vector dt is defined as dt=yt-H[Mt(x0b)].

The convergence rate of the optimization problem depends on the condition number of the Hessian matrix . In operational data assimilation systems using atmospheric models, the dimension of the state vector is typically on the order of O(1010) or greater. This results in a background error covariance matrix B that is too large to be explicitly represented or handled directly. To address this computational challenge, operational systems commonly employ the following approach : 4δx0=UxvHδxt=Utyv. Here, Ux is a square root of the background error covariance matrix B=UxUx⊤;, and v is the new control variable in the reduced-dimension space. The initial perturbation δx0 and the observation perturbation Hδxt are projected onto a subspace covered by ensemble members using the transformation matrices Ux and Uty, respectively. The perturbation matrices Ux and Uy are defined as follows: 5Ux=1Ne-1δx(1),δx(2),⋯,δx(Ne)Uy=1Ne-1δy(1),δy(2),⋯,δy(Ne), where Ne is the ensemble size, and δx(k) and δy(k) are the kth ensemble perturbations for the model state and observation space, respectively. Perturbations in observation space are calculated using the tangent linear observation operator, where δy=Hδx. By adopting this transformation, the cost function is reformulated as follows: 6J(v)=v⊤v︸Jbackground+∑t=0τUtyv-dt⊤R-1Utyv-dt︸Jobservation. To minimize Eq. (), v must satisfy the condition (∂J/∂v)⊤=0. As a result, this approach eliminates the need for an adjoint model as all calculations occur within the subspace covered by the ensemble samples. This incremental 4DEnVar approach, combined with ensemble-based transformations, thus balances computational efficiency and the practical constraints of high-dimensional data assimilation systems. For further details on these methods, we encourage readers to review the mathematical descriptions in , , , and .

In this study, the control method based on the EnKF adopts the framework proposed in . The EnKF minimizes the following cost function to obtain the analysis state: 7Jx0=x0-x0b‾⊤Pb-1x0-x0b‾︸Jbackground+y0-Hx0⊤R-1y0-Hx0︸Jobservation. Here, x0b‾ is the ensemble mean of the background state variables, and Pb represents the background error covariance matrix. As in 4DVar, MPC and EnKF consider similar cost components, taking into account the background information and discrepancies in their respective frameworks. From a variational perspective, ensemble methods like the EnKF can be interpreted as approximating the solution to a variational cost function such as Eq. (), using ensemble statistics to represent background error covariances.

The EnKF efficiently reduces the computational cost by representing the error covariance matrix Pb statistically using ensemble members as follows : 8Pb=EET,9E=1Ne-1δx(1),…,δxNe, where E is the matrix of ensemble members, with each column representing the perturbation from the forecast state. δx(k) denotes the kth ensemble perturbations for the model state. Analytically solving the cost function in Eq. () yields the update of the ensemble mean. Unlike the variational methods discussed in Sect. , which require iterative numerical optimization to minimize their respective cost functions, EnKF does not require such iterations.

Regarding the update of ensemble members, we obtain the ensemble perturbation matrix Xa using the ensemble transform Kalman filter (ETKF; ), as follows: 10Xa=XbWa,11Wa=Ne-1P̃a1/2,12P̃a=Ne-1I+Yb⊤R-1Yb-1. Here, Xb denotes the background perturbations, and P̃a represents the analysis error covariance matrix in the transformed space. Yb represents the perturbation of the background ensemble in the observation space, and the weights Wa are then derived based on the analysis covariance. Similarly to 4DEnVar, which uses ensemble approximations to project initial and observation perturbations onto a subspace covered by ensemble members, the ETKF efficiently reduces the dimensionality of the analysis problem with ensemble-based transformations.

Sequential methods, such as EnKF, update the state estimate as new observations become available, typically using a forecast–analysis cycle. In contrast, variational methods formulate the state estimation as an optimization problem over a time window, where the model trajectory is adjusted to minimize a cost function based on observations and prior estimates.

While EnKF is effective for real-time state estimation, EnKS improves estimation accuracy further by considering observations over a time window and incorporating their influence retrospectively. In this study, we employ 4D-ETKF as our implementation of EnKS; 4D-ETKF estimates the initial state by assimilating observations distributed over a finite time window, using an ensemble-based transformation that minimizes the analysis error covariance. Unlike the original EnKS that relies on sequential updates, 4D-ETKF applies a single batch update by linearly combining ensemble perturbations, ensuring consistency and computational efficiency without the need for adjoint models. For a comprehensive explanation, please refer to and .

Variational methods and EnKF estimate the analysis state by assuming Gaussian error statistics for the background and observations and minimizing the cost functions defined in Eqs. () and (). In contrast, the PF does not assume Gaussianity or linearity but approximates the entire probability distribution of the state as a set of particles (ensembles or samples). By assigning a likelihood to each particle, PF estimates the analysis state, making it suitable for systems with strong nonlinearity and non-Gaussianity. The particle distribution plays a similar role to the error covariance matrices (B and P) used in the variational methods and EnKF. Unlike these methods, however, the PF does not explicitly calculate the error covariance; instead, the particle distribution implicitly represents the statistical properties of the background error covariance. Although the likelihood function used in the PF resembles the observation term in the cost functions of other data assimilation methods, it plays a more central and explicit role in the PF.

For each particle x(k), the likelihood is calculated as follows: 13py|x(k)∝exp⁡-12y-Hx(k)⊤R-1y-Hx(k). This calculation resembles the state deviation term Jstate in Eq. () for MPC, where posterior states are penalized based on their deviation from the reference. The likelihoods are normalized to produce the particle weights λ(k): 14λ(k)=py|x(k)∑m=1Nepy|x(m). Using the weighted particles, the PF approximates the posterior distribution (filter distribution) as follows: 15p(x|y)≈∑m=1Neλ(m)δx-x(m), where δ(x-x(k)) represents a Dirac delta function centered at particle x(k). This representation indicates that the posterior distribution is expressed as a discrete set of weighted particles. To better approximate the posterior distribution and mitigate degeneracy, where some particles have negligible weights, a resampling step is performed. During resampling, particles with higher weights are replicated, while those with lower weights are discarded, ensuring the ensemble remains focused on the most likely regions of the state space.

The PF is a method for sequentially estimating states, while the PS uses future observation data to provide more accurate state estimates. Applying the weights calculated during the filter update within a data assimilation window, the PS uses the future weights to find the smoother solution at any point throughout the window. This approach is justified by the Markov property, where the system's future evolution depends solely on its current state . By taking advantage of this feature, the smoother can produce more accurate estimates over the assimilation window by using future data and previously calculated weights.

We note that several studies propose strategies to address degeneracy and maintain particle diversity e.g.,. These differences include the resampling strategy, techniques to mitigate particle collapse, and localization to manage high-dimensional systems. The current study adopts the PF and PS algorithm based on the recently proposed PF by as it employs regularization and iterative updates to effectively address degeneracy and maintain particle diversity. For more detailed information on this approach, please refer to and .

Figure  illustrates the process of the CSE using the proposed EnMPC. The procedure consists of the following steps:

To obtain an accurate estimate of the current state of the system, we first simulate observations from the nature run (NR or the true state of the system). We then perform a conventional ensemble data assimilation using these simulated observations, which corresponds to the filter update (Fig. a). This step includes estimating unobserved state variables that are targets for the control. The outcome of this process provides the initial conditions necessary for the subsequent control step.

Here, we emphasize that, for state estimation, in the first step (Fig. a), we employ conventional ensemble data assimilation methods, corresponding to the filter update. In contrast, the second step (Fig. b) utilizes the proposed EnMPC, which is based on an ensemble smoother update, to determine the optimal control inputs. For data assimilation in the first step, we consistently use the ETKF, regardless of which ensemble smoother update method (4DEnVar, EnKS, or PS) is employed in EnMPC in the second step. This uniformity ensures that any differences in performance are solely due to the choice of method in EnMPC in the second step and not influenced by variations in the state estimation in the first step. Lastly, the current study adopts a moving-horizon window of one step. That is, regardless of the length of the prediction horizon used in EnMPC, data assimilation and control input estimation are performed at every time step in each cycle.

The current study uses the Lorenz63 model for testing the proposed control method. Although relatively simple in structure, the model is widely employed as a test bed for understanding chaotic system behavior. This study aims to demonstrate the effectiveness of EnMPC for control and parameter estimation in such chaotic systems.

The Lorenz63 model is a simplified model of atmospheric convection and is represented by the following set of ordinary differential equations with three state variables: 19dxdt=σ(y-x),dydt=x(ρ-z)-y,dzdt=xy-βz. Following , we set the system parameters σ=10, ρ=28, and β=8/3. The time step is set to Δt=0.01 (units defined arbitrarily as 1 h; see ). The Lorenz63 model is characterized by its chaotic trajectory, which oscillates around two unstable fixed points, (±72,±72,27)⊤ .

Using the Lorenz63 model, the current study investigates two scenarios for control input estimation: estimating only ux, as shown in Eq. (), and estimating all three control variables ux, uy, and uz, as shown in Eq. (): 20dxdt=σ(y-x)+ux,dydt=x(ρ-z)-y,dzdt=xy-βz, and 21dxdt=σ(y-x)+ux,dydt=x(ρ-z)-y+uy,dzdt=xy-βz+uz.

The control objective in the current study is to keep the value of x in the model positive, ensuring that the system avoids undesired negative states. Note that the control inputs are applied to the time derivatives of the state variables rather than the states themselves.

In the proposed EnMPC framework, we address control problems using two approaches: the penalty term approach and the trajectory-tracking approach. Each approach employs different methods for generating objective outputs yr and operators Hr. Throughout our experiments, we set the objective output error covariance matrix Cr, which acts as the weighting matrix for the deviations between state variables and control objectives, to Cr=0.01I, where I is the identity matrix. This configuration is based on insights from preliminary experiments and the detailed investigation in .

In the penalty term approach, we restrict x to positive values by imposing penalties on regions where x≤0. Specifically, we utilize an objective output yr=0 and a control operator Hr(x)=log⁡(1+exp⁡(-ax))/a with a=0.5. In this case, we apply the control only through ux using Eq. ().

As shown in Fig. a, while x fluctuates between positive and negative values in the NR, all four MPC methods generally restrict x to the x>0 region. This demonstrates that the proposed method successfully solves the MPC problem using ensemble approximations. In addition, the penalty term approach achieves control that takes into account constraint conditions by using the objective output operator.

The comparison of control inputs ux shown in Fig. e shows that, during the initial 400 steps, the control input for EnMPC based on PS is larger than those for the other methods (4DEnVar and EnKS). As described in Sect. , this is because EnKS-based and 4DEnVar-based EnMPC use ensemble-based linear transformations, which help retain the statistical structure of the original ensemble . Specifically, when the cost function includes a penalty term weighted by the inverse of the ensemble covariance, the solution is guided toward regions of high ensemble density. This acts as a form of regularization, effectively constraining the solution to subspaces covered by the dominant ensemble modes and scaling it according to ensemble uncertainty . Compared to approaches that do not explicitly incorporate such statistical information, this often results in smaller and more dynamically consistent control inputs.

In contrast, PS-based EnMPC determines the analysis state through resampling, where particles with higher weights are replicated, while those with lower weights are removed. This can lead to the analysis state being dominated by a few specific particles, potentially causing more abrupt changes in the control input. However, this experiment uses a nonlinear observation operator Hr(x)=log⁡(1+exp⁡(-ax))/a as the penalty function, which posed challenges for EnKS-based and EnVar-based EnMPC as they inherently assume Gaussianity. On the other hand, PS-based EnMPC is more appropriate for handling non-Gaussian structures and is less affected by such assumptions .

Beyond step 400, the success rate of the control approaches nearly 100 % for all MPC methods, and, during this period, the magnitudes of control inputs for the three EnMPC methods show no significant differences. This suggests that the choice of data assimilation method influences the performance, especially during the initial stages.

When comparing conventional MPC and EnMPC, it becomes clear that EnMPC achieves significantly reduced control input magnitudes, which leads to smaller oscillations compared to conventional MPC. This is likely because conventional MPC uses a fixed control weight matrix Cu in Eq. (), whereas EnMPC estimates it from the analysis ensemble as Pa in Eq. ().

The trajectory-tracking approach controls the system state toward a predefined reference trajectory that satisfies x>0. We employ the trajectory data from as the reference and consider all three control variables ux, uy, and uz using Eq. ().

The results demonstrate that the proposed EnMPC can accurately follow the reference trajectory (Fig. a). In particular, 4DEnVar-based and EnKS-based EnMPC provide smooth and stable control inputs, while PS-based EnMPC requires larger control inputs (Fig. e–g). As mentioned in Sect. , this is because PS-based EnMPC uses particles to represent the distribution, whereas the other two methods use ensemble-based transformations. In terms of tracking performance, PS-based EnMPC achieves a significantly lower root mean squared error (RMSE) of 0.22 compared to 3.04 and 3.03 for 4DEnVar-based and EnKS-based EnMPC, respectively (Fig. ). This suggests that the PS-based EnMPC, known for its flexibility in handling nonlinear regimes, can more accurately represent complex behaviors like the reference trajectory. In contrast, EnKS-based and EnVar-based EnMPC struggle to properly incorporate the nonlinearities of the reference trajectory, resulting in larger RMSE values.

When compared to conventional MPC, all EnMPC methods exhibit significant advantages in both tracking performance and control efficiency. Conventional MPC shows an RMSE of 5.91 (Fig. ), which is considerably higher than that of any of the EnMPC methods, demonstrating its difficulty in accurately following the reference trajectory. As discussed in Sect. , this is likely to be due to the fixed control weight matrix Cu in conventional MPC, which limits its flexibility in adapting to the reference trajectory in the prediction horizon.

To enhance the accuracy of the control in both conventional MPC and EnMPC or to reduce the abrupt control inputs in PS-based EnMPC, improving the prediction horizon or increasing ensemble sizes would be effective. These improvements remain an important subject for future research.

This section provides a comparison of the computational time required by conventional MPC and various EnMPC methods across different prediction horizons (Tp). We perform the comparison for both the penalty term approach (Fig. a) and the trajectory-tracking approach (Fig. b). The success rate in Fig. a is computed as the proportion of time steps – excluding the initial 200 spin-up steps – during which the value of x remains positive.

In the penalty term approach (Fig. a), EnMPC methods consistently achieve high success rates (approximately 1.0) across all prediction horizons. In contrast, conventional MPC fails to control effectively when the prediction horizon is short (6 and 24 h). In terms of computational time, conventional MPC exhibits a sharp increase as Tp extends, reflecting its computational inefficiency due to the need for full-model evaluations to calculate optimal control inputs. For example, at Tp=120 h, the computational time for conventional MPC is 620 s. On the other hand, the EnMPC methods all show much lower computational times, with the PS-based approach yielding 121 s, the 4DEnVar-based approach yielding 81 s, and the EnKF-based approach being the most computationally efficient at 16 s. This is because the 4DEnVar and PS methods used in the current study require iterations to determine the optimal control inputs, whereas EnKS does not. Exploring alternative data assimilation methods to further reduce computational time remains an important future research topic.

For the trajectory-tracking approach (Fig. b), the PS-based EnMPC achieves the lowest RMSE, maintaining high control accuracy across all prediction horizons. This is because PS does not assume Gaussianity and effectively handles the nonlinear regime, making it well-suited for accurately representing complex reference trajectories. In contrast, conventional MPC exhibits significantly higher RMSE values, indicating difficulty in tracking the reference trajectory, regardless of Tp. In terms of computational time, PS-based EnMPC requires slightly higher computational costs compared to other EnMPC methods, but it remains much more efficient than conventional MPC (e.g., at Tp=120 h: conventional MPC = 651 s, 4DEnVar-based = 119 s, EnKF-based = 16 s, PS-based = 158 s). This suggests that PS-based EnMPC is a strong candidate for applications where high control accuracy is prioritized. Note that the relatively higher computational cost of PS-based EnMPC in this study is due to the iterative approach used to prevent particle degeneracy . Alternative PF or PS formulations may reduce computational costs while maintaining performance .

In summary, these results demonstrate that EnMPC outperforms conventional MPC in terms of both computational efficiency and control performance. Particularly for longer prediction horizons, EnMPC effectively limits computational cost increases while maintaining high control accuracy.