Observed Control: Integrated Estimation & Synthesis

• Observed control is a family of architectures where state estimation, filtering, and observer dynamics are integral to policy synthesis rather than being auxiliary.
• It leverages techniques like Kalman smoothers and history processes to reformulate predictive control problems into estimation frameworks, ensuring scalability and adaptive horizon selection.
• The framework supports observer-based implementations that combine controlled sensing and formal guarantees to address noise, partial observability, and adversarial disturbances.

Observed control can be understood as a family of control formulations in which observation, estimation, or observer dynamics are integral to policy synthesis rather than merely auxiliary to it. In one explicit usage, “Observed Control” denotes a predictive controller that turns predictive control into a state-estimation problem in reverse, uses Kalman smoothers as the backend optimization framework, and provides linear time-horizon length scalability, adaptive time horizon lengths, and early optimization termination criteria (Hamzezadeh et al., 18 Aug 2025). Across partially observed stochastic control more broadly, related constructions replace inaccessible state variables by filters, observer states, or history processes, so that control is designed on conditional distributions, estimated states, or information histories instead of the hidden physical state (Confortola et al., 18 Feb 2026, Yüksel, 2023, Haesaert et al., 2015).

1. Control–estimation duality and observer-centric formulations

The most direct formulation of observed control is the smoother-based predictive-control construction of (Hamzezadeh et al., 18 Aug 2025). There, the duality between state estimation and MPC is made explicit: the dynamics become the process model, the desired objective or tracking error becomes a measurement residual, and the control inputs are embedded into an augmented “state” that the smoother estimates. For linear systems, the augmented state

$$\chi_k=\begin{bmatrix} x_k \ u_k \end{bmatrix},\qquad \Phi=\begin{bmatrix} A & B \ 0 & \mathbb{I}_m \end{bmatrix},\qquad H=\begin{bmatrix} C & D \ 0 & \mathbb{I}_m \end{bmatrix}$$

allows the observed-control smoother objective to become equivalent to the LQR objective in the infinite-horizon limit. The paper’s key algorithmic refinement is that MPC applies only the first control, so the backward RTS pass can be eliminated in favor of a forward-only accumulation. This yields adaptive horizon selection through the termination metrics $\rho_N$ and $\tau_k$, and a decomposition of linear MPC into purely reactive and anticipatory components that enables any-time any-horizon observed control while ensuring controller stability for short time horizons (Hamzezadeh et al., 18 Aug 2025).

A related observer-centric perspective appears in observer-based realization. For the linear system $\dot x(t)=Ax(t)+Bu(t)$, $y(t)=Hx(t)$, the associated state-observer system is $\dot y(t)=HAx(t)+HBu(t)$. The observer-based realization framework treats this as a dynamical system on observables and uses bridge matrices together with dimension-keeping semi-tensor product machinery to construct approximate or exact observer dynamics. Exact closure exists if and only if the observer subspace $H_*=\operatorname{Span}(\operatorname{Row}(H))$ is $A$-invariant, equivalently $HA=\Xi H$. When exact closure fails, the framework enlarges the observer set to the smallest $A$-invariant or $\rho_N$0-invariant closure containing $\rho_N$1, producing an extended OR-system or feedback extended OR-system (Cheng et al., 2024).

2. Partial observation, filtering, and separation

The main mathematical substrate for observed control is partially observed stochastic control. In a finite-state continuous-time Markov chain with hidden state $\rho_N$2, control-dependent jump intensities $\rho_N$3, and observation process

$\rho_N$4

admissible controls are $\rho_N$5-predictable. A Girsanov-type change of measure introduces the unnormalized filter $\rho_N$6, which satisfies the Wonham/Zakai equation and becomes the fully observed state of a separated control problem. The original weak control problem and the separated problem are equivalent. For the separated problem, the value function is characterized as the unique bounded viscosity solution of the elliptic HJB equation in the infinite-horizon case and the unique viscosity solution of the parabolic HJB equation in the class of functions with linear growth in $\rho_N$7, uniformly in $\rho_N$8; the same treatment also yields verification theorems and a stochastic maximum principle (Confortola et al., 18 Feb 2026).

Belief-state reduction is not the only route. A direct history-space formulation replaces the belief-MDP by the infinite-past state

$\rho_N$9

for discounted cost and by

$\tau_k$0

for average cost. Under weak continuity of the transition and measurement kernels together with a filter/history continuity condition, optimal discounted-cost and average-cost policies exist. For discounted control, $\tau_k$1-memory policies are asymptotically optimal, and under Wasserstein regularity the finite-window approximation error is explicitly bounded, with quantization diameter upper bounded by $\tau_k$2 (Yüksel, 2023).

Risk-sensitive and jump-driven extensions preserve the same structural pattern. Partially observed forward-backward systems with Brownian motion and Poisson random measures are transformed by likelihood-ratio or Girsanov changes of measure into complete-information recursive control problems, leading to modified Zakai equations and global stochastic maximum principles even when the control domain is not convex and the control enters diffusion and jump coefficients (Lin et al., 7 Apr 2025, Zheng et al., 2021).

3. Scalable computation under partial observation

For large-scale partially observed systems, especially PDE-governed models, observed-control ideas are deployed through separation-based computation. One architecture first solves an offline open-loop deterministic trajectory optimization in belief space using a black-box simulator, then identifies a trajectory-dependent linearized reduced-order model from input-output impulse responses via time-varying Eigensystem Realization Algorithm, and finally designs an online LQG controller on the reduced model. The central justification is nominal-cost dominance: when the cost is linearized around a nominal trajectory, the first-order error satisfies $\tau_k$3, so nominal planning and local feedback may be designed separately. In the nonlinear heat example, the paper reports that the reduced-order design lowers complexity by about $\tau_k$4 compared with DDP-based RL (Yu et al., 2017).

For partially observed SPDEs, a more direct optimization route combines finite-element / implicit-Euler discretization, particle filtering, and stochastic gradient descent. The admissible controls are $\tau_k$5-progressively measurable, the conditional law $\tau_k$6 is approximated by a particle system, and the FE-PF-SGD algorithm alternates particle propagation, forward SPDE solves, backward adjoint solves, stochastic-gradient control updates, and bootstrap resampling. The paper proves spatial convergence of the finite element approximation of the forward-backward system, while explicitly stating that a full convergence theorem for the SGD/filtering-based control iteration is left to future work (Bao et al., 1 Apr 2025).

Online partially observed control with adversarial disturbances has also been studied for linear dynamical systems with hidden state $\tau_k$7, observation $\tau_k$8, and adaptive adversarial convex costs. Double Spectral Control constructs a natural observation sequence $\tau_k$9 with $\dot x(t)=Ax(t)+Bu(t)$0, then applies a two-level spectral approximation based on a universal basis of Hankel eigenvectors. The resulting improper-learning controller attains regret

$\dot x(t)=Ax(t)+Bu(t)$1

while each round can be implemented in $\dot x(t)=Ax(t)+Bu(t)$2 amortized time (Brahmbhatt et al., 27 May 2025).

4. Observer-based implementation and formal guarantees

Observer-based synthesis gives observed control a formal refinement layer. For the partially observable LTI plant

$\dot x(t)=Ax(t)+Bu(t)$3

a Luenberger observer

$\dot x(t)=Ax(t)+Bu(t)$4

is combined with a sensor-based interface

$\dot x(t)=Ax(t)+Bu(t)$5

The mismatch between the ideal state-based controller and the observed implementation is quantified by a second-order bound $\dot x(t)=Ax(t)+Bu(t)$6, yielding interface precision

$\dot x(t)=Ax(t)+Bu(t)$7

In the stochastic extension, the same construction yields approximate bisimulation and a quantified robustness bound incorporating both process-noise and measurement-noise terms (Haesaert et al., 2015).

Layered control extends this viewpoint to inter-layer imitation under partial observation. Each layer has its own observer, and the lower-layer controller is built as

$\dot x(t)=Ax(t)+Bu(t)$8

The core notion is a stochastic simulation function $\dot x(t)=Ax(t)+Bu(t)$9 that must upper-bound the mean-squared output mismatch and contract in expectation. For linear systems with steady-state Kalman estimators, this yields the explicit uniform output-distance guarantee

$y(t)=Hx(t)$0

The aerial-robotics examples report $y(t)=Hx(t)$1 for the UAV case and $y(t)=Hx(t)$2 for the hexacopter-with-camera case (Stamouli et al., 13 Apr 2026).

5. Informative actions, controlled sensing, and adversarial observation design

Observed control does not only use measurements; it can also shape what becomes measurable. One formulation trains policies whose actions themselves reveal agent state. In an aircraft target-tracking problem, a neural policy is trained with Soft Actor-Critic while an embedded Unscented Kalman Filter uses pseudocontrol actions as measurements. Reward shaping augments the distance-based task objective by an estimator-performance term, with final reported scales $y(t)=Hx(t)$3 and $y(t)=Hx(t)$4. Over 14,500 Monte Carlo episodes, the embedded-estimator policy has task performance very similar to the task-only policy at the 80th percentile, with reward $y(t)=Hx(t)$5 versus $y(t)=Hx(t)$6, but far better state reconstruction: mean end-of-episode position error is $y(t)=Hx(t)$7 m versus $y(t)=Hx(t)$8 m, mean velocity error is $y(t)=Hx(t)$9 m/s versus $\dot y(t)=HAx(t)+HBu(t)$0 m/s, less than $\dot y(t)=HAx(t)+HBu(t)$1 m position error occurs in about $\dot y(t)=HAx(t)+HBu(t)$2 of cases versus about $\dot y(t)=HAx(t)+HBu(t)$3, and the local linearized observability matrix remains singular at any single point so the improvement arises from the stripped observability matrix over trajectories rather than from pointwise full observability (Fernandez et al., 25 Jun 2026).

A more structural version of observation design appears in path integral control with controlled sensing. For

$\dot y(t)=HAx(t)+HBu(t)$4

the observation matrix itself is treated as a control variable. If sensing is restricted to a measurable selector from the matching set

$\dot y(t)=HAx(t)+HBu(t)$5

then the belief-space HJB can be transformed by $\dot y(t)=HAx(t)+HBu(t)$6 into a linear PDE with a Feynman–Kac representation. The paper also emphasizes the main limitations: the matching set may be empty, and the constrained optimum generally differs from the true unconstrained optimum (Das et al., 21 Apr 2026).

The same centrality of observations also appears in adversarial form. In a dynamic game formulation of gaslighting, a manipulator changes the observation density, perturbs the information-state recursion $\dot y(t)=HAx(t)+HBu(t)$7, and is constrained by Expected Stage-wise Stealthiness. The resulting analysis gives explicit bounds on the deviation of the information state and on degradation of the decision-maker’s objective, with cumulative effects over the horizon even when the manipulations remain stealthy (Liu et al., 2023).

6. Applications, limitations, and conceptual boundaries

Applications are diverse. The named Observed Control method is demonstrated on a mass-spring-damper LQR benchmark, linear drag with obstacle avoidance, and cart-pole swing-up; for the nonlinear benchmark, EKF- and UKF-backed observed control produce effectively indistinguishable trajectories (Hamzezadeh et al., 18 Aug 2025). Observer-based correct-by-design synthesis is evaluated on a smart-building temperature-control problem with limited sensing due to a sensor fault (Haesaert et al., 2015). Layered partially observed control is demonstrated on an unmanned aerial vehicle and a hexacopter with a camera payload (Stamouli et al., 13 Apr 2026), and partially observed Stackelberg stochastic differential games are applied to multi-agent formation control with one leader and multiple followers (Li et al., 2024).

Digital twin control offers another application domain. There, a controlled physical twin is only partially observed, while a digital twin combines ensemble Kalman filtering with a forward-in-time McKean–Pontryagin control law so that state assimilation and control computation occur simultaneously. Numerical studies on a controlled Lorenz-63 system and an inverted pendulum show bounded RMSE in the former and stabilization of the upright unstable equilibrium in the latter, even with small ensembles, but the paper explicitly describes the contribution as methodological and numerical and leaves mathematical investigation of stability and convergence to future research (Opper et al., 1 Oct 2025).

A plausible implication of these results is that observed control is less a single theorem or algorithm than a recurring design stance: policy synthesis is carried out on filters, observers, history processes, or action-generated measurements rather than on the inaccessible physical state. The same literature also marks its boundaries. Exact observer-based realization may fail unless the observer subspace is invariant (Cheng et al., 2024). Trajectory-level observability may improve even when the local observability matrix remains singular (Fernandez et al., 25 Jun 2026). Some linearly solvable partially observed constructions depend on restrictive matching conditions between observation-driven diffusion and actuation authority (Das et al., 21 Apr 2026). Observed control is therefore best understood not as a replacement for estimation, but as a class of architectures in which estimation, observation design, and control law computation are structurally inseparable.