Dynamic State Predictive Control

• Dynamic State Predictive Control is an advanced method that integrates state prediction with receding-horizon optimal control to ensure robust and constraint-adherent performance.
• It leverages both model-based and data-driven approaches, using value function approximations and dynamic state unrolling for efficient, real-time adaptation.
• It achieves significant computational efficiency and robust results across nonlinear, stochastic, and hybrid systems through integrated optimization and dynamic feedback.

Dynamic State Predictive Control (DSPC) is a family of advanced control methodologies that integrate system state prediction, optimal control, and dynamic optimization in both classical control and modern data-driven or learning-based architectures. DSPC encompasses a spectrum of approaches—from explicit embedding of value functions in model predictive control (MPC) frameworks, to dynamic (often learned) state-space controllers unrolled within deep or data-driven architectures. The unifying feature is the explicit, often recursive exploitation of the system’s state evolution—predicted, optimized, or learned—in a receding-horizon or sequential fashion to achieve robust, constraint-adherent control or estimation. This article synthesizes the key theoretical and algorithmic elements of DSPC, as established in both model-based and learning-based settings.

1. Core Problem Formulation and Theoretical Foundations

DSPC formulations universally rest upon finite-horizon optimal control of discrete- or continuous-time systems. For a general nonlinear system

$$x_{k+1} = f(x_k, u_k), \quad x_k \in \mathbb{R}^n,\quad u_k \in \mathbb{U} \subset \mathbb{R}^m,$$

the canonical receding-horizon (MPC) problem is

$$\min_{U_k \in \mathbb{U}^N} J_N(x_k,U_k) \quad \text{s.t. } x_{l+1|k} = f(x_{l|k}, u_{l|k}),$$

where

$$J_N(x_k, U_k) = x_{k+N|k}^T Q_N x_{k+N|k} + \sum_{l=k}^{k+N-1} \left[ x_{l|k}^T Q x_{l|k} + u_{l|k}^T R u_{l|k} \right].$$

Only the first input $u_{k|k}^*$ is applied; at the next sampling step, the problem is re-solved. DSPC distinguishes itself by either embedding a dynamic, quadratic approximation of the cost-to-go (value function) or by unrolling a learned or approximate state-propagation operator—thus “predictively” steering the full, instantaneous state for optimal regulation, tracking, or estimation (Chacko et al., 2023).

In continuous-time, the optimal control problem can equivalently be embedded directly into a primal-dual flow, where the controller’s internal states implement the KKT conditions of the finite-horizon OCP, leading to a dynamic system of the form

$$\begin{cases} \dot{z} = -\alpha\,\nabla_z L(z,\lambda,\mu), \ \dot{\lambda} = \alpha\,(Gz - g(\xi)), \ \dot{\mu} = \alpha\bigl[ h(z) - P_{N_+}(h(z),\mu) \bigr], \ \nu = \bar{\nu}_r + z_0. \end{cases}$$

The plant and controller are interconnected, and robust (input-to-state) stability is obtained if their respective dynamical response speeds are suitably separated (Nicotra et al., 2017).

2. Value Function Approximation and ADP-MPC Variants

A distinctive hallmark of model-based DSPC is the exploitation of approximate dynamic programming (ADP) principles within MPC schemes. For nonlinear systems, the exact value function is intractable; thus, a quadratic approximation

$$\hat{V}_{k+1}(x) = \min_{i=1,\ldots,\mu} \bar{x}_{k+1}^T \bar{P}_i \bar{x}_{k+1},$$

where $\bar{x}$ is the augmented state and $\{\bar{P}_i\}$ are precomputed via Riccati recursions over switched affine models, enables embedding dynamic programming look-ahead into the MPC law. The controller conducts a discrete search over a quantized input set to find

$$v_k^* = \arg\min_{v \in \{v_1,\ldots,v_M\}} \left\{x_k^T Q x_k + v^T R v + \hat{V}_{k+1}(f(x_k, v))\right\},$$

optionally refining this with a local search around $v_k^*$ for improved accuracy (Chacko et al., 2023). State or control constraints are incorporated either via direct pruning of the basis $\{\bar{P}_i\}$ to admissible regions (polytopic $\mathbb{X}$) or by adding penalty/barrier terms.

Experimentally, such ADP–MPC variants achieve comparable tracking and constraint adherence as full nonlinear MPC at 10–100$\times$ lower online computational cost, making them particularly attractive for fast-sampling and embedded applications.

3. Dynamic Learning and Nonlinear State-Space Unrolling

Contemporary DSPC in learning-based settings dispenses with fixed, model-based operators in favor of networks or adaptive mechanisms that dynamically generate or adjust the state-transition and readout matrices. In the MambaX nonlinear State Predictive Control framework for image super-resolution:

• The state-propagation is governed by discretized, time-varying operators:

$$h_i = \bar{A}_i h_{i-1} + B_i u_i, \quad y_i = C_i h_i + D u_i,$$

where $\bar{A}_i$, $B_i$, $C_i$ are dynamically synthesized from the current input image via parameterized MLPs or CNNs.

• The system learns the differential coefficients (e.g., $A$, $B$, $\Delta_i$) and their nonlinear, sample-dependent maps via back-propagation over reconstruction or task-specific loss functions.
• Additional mechanisms, such as state cross-control for multimodal fusion and progressive transitional (domain-crossing) learning, inject dynamic adaptability to both the system’s state update and its multimodal inputs (Li et al., 22 Nov 2025).

This approach generalizes classical state-space MPC: rather than a fixed sequence of Riccati-based look-ahead, the entire prediction and control law is orchestrated by a learned, high-dimensional, time- and input-dependent operator. This confers unique advantages for complex signal domains (e.g., hyperspectral image SR), where error propagation and domain heterogeneity must be controlled at each step of the forward process.

4. Data-Driven and Tube-Based DSPC

For LTI or stochastic systems with partially or fully unknown models, DSPC utilizes data-driven system identification and uncertainty handling:

• An observed state is decomposed as $x_k = z_k + e_k$, where $z_k$ is a nominal state predicted by data-driven models (e.g., via Hankel/Willems’ lemma from behavioral systems theory), and $e_k$ is a stochastic error driven by disturbances.
• The receding-horizon predictive control problem enforces tightened state/input tubes:

$$z_{l|k} \in \mathcal{X}_l := \{ z \mid G_x z \le \eta_l \}, \qquad \tilde{z}_{l|k} \in \mathcal{Z}_l := \mathcal{X}_l \ominus \mathcal{E}_{\mu,l}$$

where $\eta_l$ and $\mathcal{E}_{\mu,l}$ are derived via scenario-based quantile analysis using offline disturbance sequences and robustification against measurement noise (Kerz et al., 2021).

• The optimizer runs over the nominal state, with feedback gains and contraction properties ensuring recursive feasibility and input-to-state stability (ISS).

This setup provides probabilistic constraint satisfaction and robustness guarantees without requiring a parametric plant model, thus broadening the applicability of dynamic-state predictive laws in empirical regimes.

5. Dynamic Reference Tracking and Terminal Ingredient Design

DSPC settings frequently address tracking of dynamic (possibly unreachable) references. For general nonlinear plants:

• Tracking MPC augments the cost by stage and terminal penalties on the deviation from state and input references:

$$\ell(x,u,r) = \|x - x^r\|^2_Q + \|u - u^r\|^2_R, \quad V_f(x, r)$$

• Recursive feasibility and stability are achieved by the design of a parameterized terminal set $\mathcal{X}_f(r)$ and feedback law $u = k_f(x, r)$ satisfying decrease conditions:

$$V_f(x^+, r^+) \le V_f(x, r) - \ell(x, k_f(x, r), r)$$

• For periodic or dynamically unreachable references, a layered optimization structure decouples reference trajectory planning and short-horizon tracking. Online adaptation of the terminal set size is used to balance convergence rate against constraint proximity and enlarge the region of attraction (Köhler et al., 2019).

6. Extensions: Fuzzy Systems, Stochastic Jump Models, and Embedded Implementations

DSPC formalisms extend naturally to hybrid and stochastic system classes:

• In T-S fuzzy Markovian jump systems, dynamic-prediction optimization (DPO-MPC) introduces an explicit controller state $\eta_k$ and perturbation variable $c_k$; the former evolves via dynamic matrices $\mathcal{A}$, $\mathcal{C}$ determined offline, while online optimization solves a problem only over $\eta_k$ for computational efficiency. This separation ensures a large initial feasible set and mean-square stability in the presence of stochastic regime switching (Zhang, 27 Aug 2024).
• In continuous-time settings, the optimal predictive law can be implemented as a dynamical system running in parallel with the plant. When augmented with an explicit reference governor, the set of admissible initial states is substantially enlarged, and asymptotic convergence to non-steady references is achieved without constraint violations (Nicotra et al., 2017).
• Low-dimensional dynamic state modeling (e.g., via dynamic mode decomposition) can be fused with image data for scalable prediction and control of spatially extended physical processes, supporting real-time implementation even in high-dimensional state spaces (Lu et al., 2020).

7. Empirical Performance and Implementation Considerations

Across regimes (nonlinear, data-driven, learning-based, stochastic), DSPC schemes consistently demonstrate:

• Orders-of-magnitude reduction in online computation compared to full nonlinear MPC (e.g., 100$\times$ CPU reduction with comparable regulation quality in ADP–MPC (Chacko et al., 2023)).
• Reduction of constraint violations and improved handling of error propagation in dynamic, high-dimensional applications (e.g., 0.3–0.8 dB PSNR gains in MambaX super-resolution on hyperspectral and pansharpening tasks (Li et al., 22 Nov 2025)).
• Scalable feasibility sets and robust performance under regime uncertainty in hybrid/fuzzy systems (Zhang, 27 Aug 2024).
• Flexible, recursive feasibility through dynamic terminal set optimization and explicit reference management (Köhler et al., 2019).

Implementation typically involves an offline synthesis step to precompute dynamic or feedback gains, quadratic cost approximations, or reduced-order models, followed by a lightweight online loop focused on prediction updates, minimum search or small-scale optimization, and immediate state feedback.

Table: Illustrative DSPC Approaches and Core Elements

Approach / Paper	Key Features	Domain / Application
ADP–MPC (Chacko et al., 2023)	Switched-system Riccati-based value approximation, fast online search	Nonlinear tank regulation
Dynamic nSPC (MambaX) (Li et al., 22 Nov 2025)	End-to-end learned dynamic state-space, cross-control fusion	Multimodal image super-resolution
Tube-based DD-SMPC (Kerz et al., 2021)	Nominal/error decomposition, chance-constrained tube tightening	Data-driven LTI with disturbances
DPO–MPC (Zhang, 27 Aug 2024)	Dynamic feedback state, perturbation augmentation	Fuzzy Markov jump systems
Continuous-time DSPC (Nicotra et al., 2017)	Primal-dual OCP flow, explicit reference governor	Embedded/real-time stabilization

This table summarizes dominant DSPC design principles across key recent literature.

• (Chacko et al., 2023) Approximate Dynamic Programming based Model Predictive Control of Nonlinear systems
• (Li et al., 22 Nov 2025) MambaX: Image Super-Resolution with State Predictive Control
• (Kerz et al., 2021) Data-driven tube-based stochastic predictive control
• (Nicotra et al., 2017) Embedding Constrained Model Predictive Control in a Continuous-Time Dynamic Feedback
• (Köhler et al., 2019) A nonlinear tracking model predictive control scheme for dynamic target signals
• (Zhang, 27 Aug 2024) Model Predictive Control for T-S Fuzzy Markovian Jump Systems Using Dynamic Prediction Optimization
• (Lu et al., 2020) Image-Based Model Predictive Control via Dynamic Mode Decomposition

These works collectively establish DSPC as a comprehensive paradigm for state- and prediction-centric control synthesis in both conventional and learning-driven domains, offering a spectrum of tradeoffs in accuracy, computational tractability, robustness, and adaptability.