Continuous-Discrete Kalman Filter (CD-KF)

• Continuous-discrete Kalman filtering is a framework that estimates continuous-time states using recursive prediction and discrete measurement corrections.
• It alternates between analytically or numerically propagating state moments over time and updating the estimates when new, discretely sampled data arrive.
• Recent advances integrate Gaussian closures, sigma-point methods, and robust numerical integration to enhance stability and accuracy in diverse applications.

Continuous-discrete Kalman filtering denotes recursive state estimation for systems whose latent dynamics evolve in continuous time while measurements are available only at discrete instants. In the formulations treated in recent work, the hidden process is specified either by a linear Itô model or by a nonlinear stochastic differential equation (SDE), and the estimator alternates between a continuous-time prediction step on each interval $[t_{k-1},t_k]$ and a discrete measurement correction at $t_k$. In the linear-Gaussian case this is the standard continuous-discrete Kalman filter (CD-KF); in nonlinear settings it becomes a family of Gaussian-assumed filters, including continuous-discrete extended, unscented, cubature, hybrid extended-cubature, and level-set variants, together with square-root and SVD factored implementations designed to improve numerical robustness (Ahdab et al., 15 Jul 2025, Nielsen et al., 2022, Kulikova et al., 2023).

1. Canonical state-space formulation

A standard CD-KF model couples a continuous-time state equation with a discrete-time observation equation. A linear form used in recent scheduling and learning work is

$$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$

with measurements at times $t_i$,

$$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$

A nonlinear form used in continuous-discrete estimation studies is

$$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$

$y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$

with Gaussian initial state and Gaussian measurement noise (Ahdab et al., 15 Jul 2025, Nielsen et al., 2022).

In this setting, “continuous-discrete” means that the latent trajectory is not updated by a fixed discrete transition alone. Instead, the filter must propagate posterior information through continuous-time dynamics between observation times and then apply a discrete correction when data arrive. For nonlinear diffusions this propagation can also be interpreted through the Fokker–Planck equation. The level-set formulation, for example, starts from

$$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$

and approximates the density by a Gaussian

$u(x,t)\approx \mathcal{N}(\bar{x}(t),\Sigma(t)),$

so that only the mean and covariance, or a square-root covariance factor, are propagated (Wang et al., 2021).

The core structural distinction from purely discrete Kalman filtering is therefore not the algebra of the measurement correction but the presence of a continuous-time prediction subproblem. That subproblem may be solved analytically in linear cases, by moment ODEs in extended filters, by sigma-point or cubature constructions, or by other Gaussian closure schemes.

2. Linear-Gaussian recursion and its sampled-data interpretation

For the linear-Gaussian CD-KF, the prediction step evolves the conditional mean and covariance continuously:

$$\frac{d\mu}{dt}=A(t)\mu,\qquad \frac{d\Sigma}{dt}=A(t)\Sigma+\Sigma A^\top(t)+\sigma(t)\sigma^\top(t).$$

At each measurement time $t_k$0, the correction is

$t_k$1

$t_k$2

with

$t_k$3

The covariance evolution can also be written in an impulse form,

$t_k$4

which makes explicit that uncertainty grows continuously and is reduced only at discrete sensing times (Ahdab et al., 15 Jul 2025).

A closely related nonlinear-EKF form propagates

$t_k$5

$t_k$6

where $t_k$7 is the drift Jacobian along the current estimate. At $t_k$8, the innovation, innovation covariance, and Kalman gain are

$t_k$9

$$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$0

followed by

$$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$1

and a stabilized covariance update (Nielsen et al., 2022).

The sampled-data interpretation of CD-KF is sharpened by convergence results. For a continuous-time linear stochastic system, if one refines the temporal discretization of the outputs and runs a discrete-time Kalman filter on those denser samples, the estimate converges to the continuous-time Kalman–Bucy estimate. The literature establishes almost sure and mean-square convergence and provides explicit rate bounds, including finite-dimensional, infinite-dimensional, bounded-observation, and unbounded-observation regimes [(Aalto, 2014); (Aalto, 2015)].

3. Nonlinear Gaussian-assumed extensions

In nonlinear continuous-discrete filtering, the main design choice is how to approximate the prediction and measurement moments under nonlinearity. The continuous-discrete EKF uses local linearization in the time update, with the covariance ODE driven by the Jacobian of the drift and the measurement update based on linearization of $$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$2 (Nielsen et al., 2022).

The continuous-discrete cubature Kalman filter (CD-CKF) replaces exact Gaussian integrals by the third-degree spherical-radial cubature rule. With cubature points

$$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$3

and a covariance square root $$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$4, the state-space points are

$$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$5

The filter uses these points in continuous-time propagation and in the discrete measurement update, from which it forms the predicted measurement, residual covariance, cross-covariance, Kalman gain, and posterior moments. In the implementations discussed in the literature, the continuous-time SDE is usually approximated on substeps by either Euler–Maruyama of order $$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$6 or Itô–Taylor of order $$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$7 (Kulikova et al., 2024).

A hybrid variant, the continuous-discrete extended-cubature Kalman filter (CD-ECKF), combines EKF-style continuous-time moment propagation with CKF-style measurement correction. Its time update solves

$$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$8

with $$dx = A(t)x\,dt + \sigma(t)\,dW,\qquad x(0)\sim \mathcal N(\mu_0,\Sigma_0),$$9, while the measurement update uses cubature points and the standard CKF formulas for $t_i$0, $t_i$1, $t_i$2, $t_i$3, and $t_i$4 (Kulikova et al., 2023).

The continuous-discrete UKF occupies a different point in the design space. One formulation propagates moment differential equations,

$t_i$5

$t_i$6

while another propagates sigma-point differential equations directly through a larger ODE system. The measurement update remains the standard unscented innovation correction (Kulikova et al., 2023).

The level set Kalman filter (LSKF) modifies the time update more radically. Instead of using an Itô–Taylor expansion and derivative-based moment formulas, it derives an ODE from the motion of Gaussian level sets under the Fokker–Planck equation. In its averaged-velocity form,

$t_i$7

with $t_i$8, while the measurement update is the standard square-root cubature Kalman measurement update (Wang et al., 2021).

A further related construction is the dual Kalman-filter-like estimator (“DuKF”), which retains the continuous-discrete predict–update structure but replaces numerical propagation of the nonlinear SDE by moment evaluation through a precomputed dual birth-death process, followed by a standard Kalman correction (Ohkubo, 2014).

This variety suggests that CD-KF is better understood as a structural framework than as a single nonlinear algorithm. The common invariant is continuous-time uncertainty propagation with discrete-time data assimilation; the main differences lie in Gaussian closure, quadrature, linearization, and numerical realization.

4. Numerical integration, discretization control, and square-root stabilization

A recurrent theme in recent CD-KF work is that filtering accuracy depends strongly on the numerical treatment of the continuous-time prediction step. In the hybrid CD-ECKF, the state mean and covariance are propagated by ODEs, and the filter is described as accurate only when these moment differential equations are solved with negligible numerical error. The paper therefore uses an adaptive variable-step Nested Implicit Runge–Kutta pair with built-in local and global error control, and emphasizes that fixed-step Euler–Maruyama or Itô–Taylor implementations do not offer discretization error control and can degrade badly for long or irregular sampling intervals, missed measurements, or stiff dynamics (Kulikova et al., 2023).

The same point appears in the continuous-discrete UKF literature. There the recommended implementation strategy is to convert the propagation step to ODEs and hand them to MATLAB adaptive solvers such as ode45, with options of the form $u(x,t)\approx \mathcal{N}(\bar{x}(t),\Sigma(t)),$6 so that local error remains below a user-specified tolerance. This removes the need for a prefixed equidistant mesh and is described as especially useful for irregular sampling (Kulikova et al., 2023).

Numerical robustness of the covariance recursion is a second central issue. One stabilization route is the Joseph form,

$t_i$9

used explicitly in a continuous-discrete EKF implementation for numerical stability (Nielsen et al., 2022). A stronger route is to propagate a square-root factor rather than the covariance itself. In the SVD-based formulations,

$$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$0

and the filter updates $$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$1 and $$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$2 by SVDs of pre-arrays assembled from propagated covariance factors, process-noise factors, or measurement arrays. This preserves symmetry and positive semidefiniteness and is reported to be more robust under finite precision arithmetic than direct covariance manipulation (Kulikova et al., 2023, Kulikova et al., 2024).

For UKF-type square-root filters, the literature distinguishes pseudo-square-root methods based on Cholesky factorization plus one-rank cholupdate downdates from “true” square-root methods based on $$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$3-orthogonal or hyperbolic QR transformations. The latter are introduced specifically to handle indefinite quadratic forms that arise when sigma-point weights are negative, avoiding repeated downdates that may fail when the downdated matrix is not positive definite (Kulikova et al., 2023).

An important conceptual clarification is that square-root, factored, or SVD implementations are not new filtering principles. They are alternative realizations of the same CD-KF recursion, introduced to improve roundoff stability, preserve covariance structure, and make nearly singular or ill-conditioned cases computationally safer (Kulikova et al., 2023, Kulikova et al., 2024).

5. Scalability, quasi-continuous assimilation, and resource-aware scheduling

One branch of the literature addresses the regime of high-frequency or quasi-continuous data assimilation. The hierarchical-matrix Kalman filter (HiKF) considers the discrete-time linear Gaussian model

$$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$4

and specializes to a random walk,

$$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$5

with $$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$6, justified by the statement that when data are acquired in rapid succession, “changes between subsequent states are small.” HiKF reformulates the Kalman recursion in terms of the cross-covariance $$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$7 rather than the full covariance $$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$8, and uses $$y(t_i)=C(t_i)x(t_i)+v(t_i),\qquad v(t_i)\sim \mathcal N(0,R(t_i)).$$9-matrix structure to compute $$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$0 in $$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$1 instead of $$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$2 for $$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$3 (Li et al., 2014).

This is not a general continuous-time state equation in the classical CD-KF form; it is described instead as a quasi-continuous discrete-time assimilation filter. Within that regime, the computational gains reported are substantial. For a $$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$4 grid and 41 assimilation steps over 5 days with surveys every 3 hours, the reported runtimes are 2.25 minutes for HiKF, 263.7 minutes for KF, and 16.3 minutes for EnKF; the corresponding storage figures are 117 MB, 21125.1 MB, and 230.5 MB. The paper also reports that EnKF’s variance can be about 1000 times larger than KF in the example, whereas HiKF reproduces the KF/LMMSE solution very accurately (Li et al., 2014).

A more explicitly continuous-discrete extension concerns sensor scheduling and selection with auxiliary dynamics. In a multi-sensor CD-KF, each sensor $$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$5 may have its own measurement model

$$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$6

with measurements arriving irregularly and asynchronously. Measurement occurrences are modeled as independent inhomogeneous Poisson processes with rates $$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$7, which leads to the mean-field covariance dynamics

$$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$8

The paper derives an upper bound on the mean posterior covariance matrix along the mean auxiliary state, proves that the bound dominates the true mean posterior covariance in Loewner order, and exploits differentiability with respect to the rates to pose a finite-horizon optimal control problem over sensor rates and auxiliary controls (Ahdab et al., 15 Jul 2025).

After optimization, actual sensing times are not drawn by simulating Poisson events; instead they are constructed deterministically by quantizing the normalized intensity measure using conditional centroids. Reported applications include robot sensing with energy state, radiation damage, water-quality monitoring with fouling and active defouling, spacecraft monitoring with time-varying visibility and solar-dependent sensor quality, and receding-horizon target tracking. In these examples, measurement decisions are themselves part of the system dynamics, which extends CD-KF from passive estimation to estimation under explicit operational constraints (Ahdab et al., 15 Jul 2025).

6. Smoothing, posterior computation, and parameter learning

CD-KF also provides the inferential core for continuous-time system identification from irregularly sampled measurements. In a continuous-time linear latent model,

$$dx(t)=f\!\big(t,x(t),u(t),d(t),\theta\big)\,dt+\sigma\!\big(t,x(t),u(t),d(t),\theta\big)\,d\omega(t),$$9

with irregular intervals $y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$0, the forward prediction uses

$y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$1

$y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$2

where

$y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$3

Thus the process covariance accumulated between observations is interval-dependent rather than fixed (Halmos et al., 2023).

A notable development is an analytical two-filter posterior representation,

$y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$4

together with a backward likelihood recursion that yields a continuous-time analog of a Kalman backward pass. The resulting smoothed mean and covariance are

$y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$5

$y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$6

and are proved equivalent to Rauch–Tung–Striebel smoothing in first and second moments (Halmos et al., 2023).

These smoothed sufficient statistics support an EM procedure for estimating the continuous-time parameters $y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$7 and $y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$8, as well as $y(t_k)=h\!\big(t_k,x(t_k),\theta\big)+v(t_k),$9, $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$0, $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$1, and $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$2. Because the formulation is continuous-time, missing observations or irregular sampling are handled simply by larger values of $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$3, without forcing the data onto a regular grid. The reported comparison shows that the continuous-time method remains approximately stable as step-size irregularity increases, and is more robust than discrete-time EM when the spectral radius of the dynamics matrix increases (Halmos et al., 2023).

A plausible implication is that CD-KF is not only a filtering device but also an estimation scaffold for continuous-time latent dynamical systems. This role becomes particularly important when discrete-time parametrizations are no longer faithful because sampling intervals vary substantially across observations.

7. Convergence theory, applications, and conceptual boundaries

The most precise asymptotic theory concerns the relation between sampled discrete-time filtering and continuous-time estimation. In finite dimensions, for a continuous-time linear stochastic system without input noise, the mean-square discrepancy between the estimate based on $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$4 samples and the continuous-time estimate satisfies an $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$5 bound. With input noise, the dominant term becomes $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$6. In infinite dimensions with bounded observation operator, the rate is $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$7 under the smooth-covariance assumption $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$8, and $$\frac{d u}{dt} = \frac{1}{2}\nabla\cdot K\nabla u - \nabla\cdot(vu),$$9 if $u(x,t)\approx \mathcal{N}(\bar{x}(t),\Sigma(t)),$0 almost surely. Without sufficient smoothness, convergence can be arbitrarily slow (Aalto, 2014).

The unbounded-observation case has also been analyzed. For systems with diagonalizable generators, admissible observation operators, or analytic semigroups, the discrete-time Kalman filter estimate converges to the continuous-time Kalman–Bucy estimate with explicit mean-square rate bounds. The paper covers pointwise and boundary measurements that arise in PDE models, including examples such as the 1D wave equation with pointwise observation, for which the discrepancy behaves like $u(x,t)\approx \mathcal{N}(\bar{x}(t),\Sigma(t)),$1, and analytic-semigroup regimes with $u(x,t)\approx \mathcal{N}(\bar{x}(t),\Sigma(t)),$2 rates; for the 1D heat equation with boundary observation, the theorem yields rates as good as $u(x,t)\approx \mathcal{N}(\bar{x}(t),\Sigma(t)),$3 for arbitrarily small $u(x,t)\approx \mathcal{N}(\bar{x}(t),\Sigma(t)),$4 (Aalto, 2015).

Reported application domains are correspondingly broad. Numerical studies include target tracking with irregular sampling and ill-conditioned measurements, gas-phase reversible reaction in a CSTR, and stochastic Van der Pol oscillators in nonstiff and stiff regimes (Kulikova et al., 2023); coordinated-turn radar tracking under nearly singular measurement covariance (Kulikova et al., 2024); modified four-tank state and disturbance estimation, where runtime and mean absolute percentage error are reported for EKF, UKF, EnKF, and PF implementations (Nielsen et al., 2022); quasi-continuous CO$u(x,t)\approx \mathcal{N}(\bar{x}(t),\Sigma(t)),$5 plume monitoring with repeated seismic data (Li et al., 2014); robot sensing with energy constraints, radiation damage, water-quality monitoring, spacecraft monitoring, and receding-horizon target tracking (Ahdab et al., 15 Jul 2025); and continuous-time latent modeling of a toggle-switch genetic circuit under irregular sampling (Halmos et al., 2023).

Several recurring misconceptions are clarified by this literature. First, CD-KF is not identical to the Kalman–Bucy filter: the former is a hybrid estimator with continuous prediction and discrete updates, whereas the latter is the continuous-time limit obtained under dense observation. Second, “CD-KF” does not name a unique nonlinear algorithm; it includes EKF-, UKF-, CKF-, cubature-measurement, level-set, and duality-based constructions that share the same continuous/discrete architecture but differ in approximation strategy [(Wang et al., 2021); (Kulikova et al., 2023); (Ohkubo, 2014)]. Third, robust performance is not determined by the Bayesian recursion alone. The literature repeatedly shows that discretization error control, covariance factorization, and conditioning of the linear algebra are often decisive, especially for stiff, irregular, long-interval, and ill-conditioned estimation problems (Kulikova et al., 2023, Kulikova et al., 2023).