Kalman Filter Divergence: Causes, Consequences, and Mitigation Strategies
I. Introduction: The Kalman Filter and the Challenge of Divergence
The Kalman filter stands as a cornerstone in modern estimation theory, providing a powerful algorithmic framework for inferring the state of dynamic systems from noisy and incomplete measurements. Its widespread adoption across diverse fields underscores its utility and theoretical elegance. However, the practical application of the Kalman filter is not without its challenges, chief among them being the phenomenon of filter divergence. This report provides an expert-level examination of Kalman filter divergence, elucidating its definition, underlying causes, and the array of strategies developed to prevent or mitigate its detrimental effects.
A. The Kalman Filter: An Optimal Recursive Estimator
The Kalman filter, also known as linear quadratic estimation (LQE), is an algorithm that processes a series of measurements observed over time, which inherently contain statistical noise and other inaccuracies. Its purpose is to produce estimates of unknown system variables that tend to be more accurate than those obtainable from a single measurement alone.1 The filter operates recursively, meaning it can process data in real-time using only the present input measurements, the previously calculated state, and its associated uncertainty matrix; no extensive history of past information is required.1
The algorithm functions through a two-phase process: prediction and update.1
1.	Prediction Phase: In this phase, the filter projects the current state estimate and its uncertainty forward in time, based on a model of the system's dynamics. This yields an a priori estimate for the next time step.
2.	Update Phase: Once a new measurement is available, the a priori estimate is refined by incorporating this new information. The update is performed using a weighted average, where more weight is given to estimates (either the prediction or the measurement) with greater certainty (i.e., smaller variance).1 The weights are determined by the Kalman gain, a crucial parameter that balances the reliance on the model prediction versus the new measurement.
Under a specific set of assumptions—namely, that the system is linear, its dynamic and measurement models are perfectly known, and all process and measurement noises are zero-mean, Gaussian, and uncorrelated—the Kalman filter is an optimal estimator in the sense that it minimizes the mean squared error of the state estimate.1 Its applications are numerous and varied, including guidance, navigation, and control of vehicles (aircraft, spacecraft, ships), time series analysis in signal processing and econometrics, robotic motion planning, and even modeling aspects of the central nervous system's control of movement.1
The celebrated optimality of the Kalman filter, however, is not an absolute guarantee of performance in all scenarios. This optimality is strictly conditional upon the aforementioned set of assumptions holding true.3 In practical engineering applications, these assumptions are often violated to varying degrees. System models are rarely perfect representations of reality; noise characteristics may not be precisely known or purely Gaussian; and underlying dynamics can exhibit non-linearities. It is precisely when these idealized conditions are not met that the filter's performance can degrade, sometimes catastrophically, leading to the problem of divergence. The filter's robust performance is thus intrinsically linked to the validity of its foundational assumptions, making divergence a common and critical concern in its practical deployment.
B. Defining Divergence: When Estimates Go Astray
Kalman filter divergence is a phenomenon characterized by a significant and persistent deviation of the filter's state estimates from the true state of the system.3 In essence, the filter "loses track" of the actual system behavior. More formally, divergence occurs when the actual estimation errors become substantially larger than the errors predicted by the filter's own internal measure of uncertainty, the state error covariance matrix, denoted as P.5 The filter might, for instance, indicate high confidence in its estimates (a small P matrix) while, in reality, the estimates are far from the true values.
This discrepancy between perceived and actual accuracy is a hallmark of divergence. The state error covariance matrix P is fundamental to the Kalman filter's operation; it quantifies the uncertainty in the state estimate and is used to compute the Kalman gain K. The Kalman gain, in turn, dictates how much weight is given to new measurements versus the model's prediction.1 If P becomes an unreliable indicator of the true error, the Kalman gain will be computed incorrectly. For example, if P becomes erroneously small, the filter becomes "smug" or overconfident in its model predictions and assigns too little weight to incoming measurements that could correct its estimate.6 This can lead to a situation where the filter effectively ignores new data, allowing its estimate to drift further from reality based on its flawed model or accumulated errors. In some cases, computational errors can even cause the P matrix to lose its essential mathematical properties, such as positive definiteness, rendering the subsequent calculations, including the Kalman gain, theoretically unsound.3 Thus, divergence is not merely about the state estimate being inaccurate; it signifies a fundamental breakdown in the filter's self-assessment of its own uncertainty, undermining the very logic of its optimal weighting scheme.
C. The Critical Need to Address Divergence
The consequences of Kalman filter divergence can be severe, particularly in applications where system state estimates are used for decision-making or control. In safety-critical systems, such as aircraft navigation 5, spacecraft control, or autonomous vehicle guidance 2, erroneous state estimates can lead to incorrect control actions, potentially resulting in mission failure, equipment damage, or even loss of life. For instance, an aerospace system relying on a divergent Kalman filter for attitude or position estimation might make incorrect thruster firings or control surface adjustments, leading to instability or deviation from the desired trajectory.5 Similarly, in robotics, inaccurate localization or perception due to filter divergence can result in collisions or failure to perform tasks. Therefore, understanding the causes of divergence and implementing effective strategies to prevent or mitigate it is of paramount importance for the reliable and safe operation of systems that depend on Kalman filtering.
II. Root Causes of Kalman Filter Divergence
The divergence of a Kalman filter can be traced back to a variety of sources, broadly classifiable into inaccuracies within the mathematical models used by the filter and numerical issues arising from its implementation on digital computers. A thorough understanding of these root causes is essential for diagnosing and addressing divergence in practical applications.
A. Model Inaccuracies and Mismatches
A primary driver of Kalman filter divergence stems from discrepancies between the mathematical model embedded within the filter and the actual physical system or environment it aims to represent.3 The Kalman filter's equations are derived based on these models, and if the models are flawed, the filter's performance will inevitably degrade.4
1. System Dynamics Misrepresentation (F matrix, dynamical biases)
Errors in the state transition model (represented by the matrix Fk​ in the discrete-time Kalman filter equations, xk​=Fk​xk−1​+Bk​uk​+wk−1​) are a significant source of divergence. These errors can arise from various factors, such as neglecting significant terms in the underlying physical equations (e.g., omitting higher-order terms in a gravitational potential model), using inaccurate values for physical constants, or failing to account for all relevant forces acting on the system.7
A particularly problematic form of system dynamics misrepresentation is the presence of "dynamical biases." These are persistent, systematic errors in the system model, such as an unmodeled constant force, an incorrect aerodynamic drag coefficient, or a persistent sensor bias that affects the system's evolution.7 For example, if an aircraft's drag is consistently underestimated in the model, the filter's prediction of its deceleration will be incorrect, leading to accumulating errors in the estimated velocity and position. Figure 1 in 7 illustrates divergence caused by a 25% error in drag acceleration.
Dynamical biases are especially insidious because their effects often compound over time. The Kalman filter is a recursive algorithm, meaning that the state estimate at each time step is based on the estimate from the previous step, propagated through the system model.1 If this model contains a persistent bias, each prediction step will incorporate and perpetuate this bias. Over numerous iterations, these small, systematic errors can accumulate, causing the filter's state estimate to drift progressively further from the true system state. This drift is exacerbated if the filter simultaneously, and incorrectly, reduces its state error covariance matrix P, thereby becoming overly confident in its flawed model. In such a state of overconfidence, the filter assigns less weight to new measurements that might otherwise correct the drift, effectively locking itself into an erroneous trajectory and accelerating the divergence.
2. Erroneous Noise Covariance Specification (Q and R, "Filter Smugness")
The process noise covariance matrix Q and the measurement noise covariance matrix R play a critical role in the Kalman filter's performance. Q accounts for uncertainties and unmodeled disturbances in the system dynamics model, while R characterizes the noise and inaccuracies in the measurements.1 Incorrect specification of these matrices is a very common and significant cause of filter divergence.6
●	If Q is chosen to be too small, the filter assumes its dynamic model is more accurate than it truly is. This leads to the filter placing excessive trust in its model predictions and insufficient weight on new measurements. This condition is often termed "filter smugness," where the filter becomes overconfident and unresponsive to data that contradicts its model.6 Such a scenario can arise if certain states or modes of the system are, in reality, subject to disturbances that are not adequately represented by the modeled process noise.5 Consequently, the filter's estimate can drift away from the true state without the measurements providing sufficient correction.
●	Conversely, if Q is too large, the filter assumes the model is highly uncertain, leading it to rely too heavily on measurements. This can make the estimates noisy and overly sensitive to measurement errors.
●	Similarly, if R is too small, the filter over-trusts the measurements, incorporating too much measurement noise into the state estimate.
●	If R is too large, the filter discounts the measurements excessively, relying too heavily on its (potentially flawed) model predictions.
The challenge of accurately specifying Q and R is often compounded by the fact that these matrices are used not only to represent actual stochastic noise but also as tuning parameters to compensate for other unmodeled dynamics or model deficiencies. For instance, the process noise covariance Q is frequently "inflated" or adjusted during a "filter tuning" process to prevent divergence.3 This injected "pseudo-noise" is not necessarily reflective of actual physical disturbances but rather serves as a mechanism to inform the filter that its model predictions should be treated with less certainty, perhaps due to unmodeled dynamics or systematic errors in the model structure itself (F or H matrices).3 Consequently, difficulties in setting appropriate values for Q and R, or the need for substantial empirical tuning, can be symptomatic of deeper, unaddressed deficiencies in the fundamental structural model of the system. The filter may appear to function with tuned Q and R, but it might be operating sub-optimally because it is effectively papering over more fundamental modeling errors.
3. Unaccounted Dynamics and State Vector Omissions
Divergence can also occur if the state vector x used in the Kalman filter does not include all the dynamic variables that significantly influence the system's behavior or the measurement process.4 If a relevant component of the system's state is omitted from the model—either deliberately to reduce computational complexity or inadvertently because its existence or relevance is unknown—the filter will be unable to correctly model or predict the system's evolution. An illustrative example provided in 7 involves estimating the altitude of a vehicle assumed to be constant. If the vehicle is actually climbing or falling at a constant rate (an unmodeled state), the filter designed for constant altitude will diverge. Since perfect mathematical modeling is often intractable for complex systems, engineers typically include only the dominant modes of behavior in the model; however, the cumulative effect of many small, unmodeled dynamics can degrade estimation accuracy and potentially lead to divergence.4
This issue highlights an inherent trade-off in defining the state vector for a Kalman filter. A more comprehensive state vector, including more variables, can potentially capture more complex system dynamics and reduce the risk of divergence due to unmodeled effects. However, a larger state vector directly translates to larger matrices (F,P,H,Q) in the Kalman filter equations. This, in turn, increases the computational burden of the prediction and update steps, as matrix operations (multiplication, inversion) scale non-linearly with dimension (e.g., matrix multiplication is typically O(n3) for n×n matrices). Engineers often face a design decision: opt for a simpler, computationally cheaper filter with a reduced state vector, which risks divergence if significant dynamics are omitted; or develop a more complex, computationally intensive filter with an augmented state vector, which may be more robust but also more challenging to implement, tune, and run in real-time. The deliberate simplification of models by omitting states is a common engineering compromise that can, if not carefully managed, lead to filter failure.7
4. The Influence of Initial Conditions (x0​,P0​)
The Kalman filter is a recursive algorithm that requires initial values for the state estimate (x0​) and the state error covariance matrix (P0​). While for observable systems, the filter's dependence on these initial conditions should theoretically diminish over time as more measurements are processed, grossly incorrect initializations can lead to transient poor performance or, in some cases, contribute to divergence.10
If the initial state estimate x0​ is far from the true initial state, and the initial error covariance P0​ is chosen to be too small (implying high confidence in an inaccurate x0​), the filter may take a very long time to converge to the true state, or it might even diverge if other model errors are present. An overly confident (small P0​) and incorrect x0​ will cause the filter to heavily discount early measurements that contradict its initial belief, thereby hindering its ability to correct itself. The optimal filtering results presuppose reasonably accurate knowledge of these initial statistics.10
The initial covariance P0​ effectively modulates the filter's initial "learning rate." A large P0​ signifies high uncertainty in the initial state estimate x0​. This makes the Kalman gain K larger in the initial steps (as K is roughly proportional to PHT(HPHT+R)−1), causing the filter to rely more heavily on the early measurements to correct its state estimate. This allows for rapid convergence if x0​ was indeed uncertain. Conversely, a small P0​ indicates high confidence in x0​. This results in smaller initial Kalman gains, making the filter more skeptical of initial measurements that deviate significantly from its predicted state (based on x0​). If x0​ is substantially incorrect and P0​ is inappropriately small, the filter will "learn" too slowly from new data. It may persistently reject valid corrective information from measurements, and if this is compounded by other issues like model mismatch or unmodeled dynamics, the filter might fail to converge or even diverge. Thus, while the filter is designed to be robust to some level of initial uncertainty, poor choices for x0​ and particularly P0​ can cripple its early performance and set it on a path towards divergence, especially in challenging scenarios.
B. Numerical Instability and Computational Pitfalls
Beyond errors in the mathematical model, divergence can also arise from the practical limitations of implementing the Kalman filter equations on digital computers with finite numerical precision.3 These issues are related to the accumulation of round-off errors and the potential loss of essential mathematical properties of key matrices, particularly the state error covariance matrix P.
1. Finite Precision Arithmetic and Round-Off Error Accumulation
Kalman filter algorithms involve numerous matrix operations (multiplication, addition, inversion) that are executed iteratively. When these computations are performed on digital computers using finite-precision arithmetic (e.g., single-precision or double-precision floating-point numbers), small round-off errors are introduced at almost every step.7 While individual round-off errors may be minuscule, their cumulative effect over many recursive cycles can become significant, leading to a gradual degradation of the filter's accuracy and, ultimately, divergence.3 This problem is particularly acute in filters that run for extended periods, systems with a wide dynamic range in their state variables, or when implemented on hardware with limited word length (e.g., older embedded systems or highly resource-constrained platforms).7
The recursive nature of the Kalman filter, a key to its computational efficiency, can inadvertently act as an amplifier for these numerical errors. Each iteration's output (the updated state estimate and covariance) becomes the input for the next. If small errors are introduced into these quantities, they are fed back into the calculations. In ill-conditioned systems or with certain forms of the update equations, these errors may not be damped out but can instead propagate and even grow with each cycle. The covariance matrix update is particularly vulnerable. This makes long-running filters or those processing data at very high rates susceptible to a slow buildup of numerical "noise" that can eventually overwhelm the true signal in the filter's internal state, leading to erroneous estimates and divergence.
2. Loss of Covariance Matrix Properties (Symmetry, Positive Definiteness)
The state error covariance matrix P is, by its theoretical definition, a symmetric and positive (semi-)definite matrix. Symmetry implies that Pij​=Pji​, and positive definiteness means that for any non-zero vector z, the quadratic form zTPz>0 (or ≥0 for positive semi-definiteness), which implies, among other things, that all its eigenvalues are positive (or non-negative). These properties are crucial for the mathematical validity and stability of the filter; for instance, variances (diagonal elements of P) cannot be negative.
However, due to the accumulation of round-off errors in finite-precision arithmetic, the computed P matrix can lose these essential properties.3
●	Loss of Symmetry: Numerical inaccuracies can lead to Pij​ no longer being equal to Pji​.
●	Loss of Positive Definiteness: More critically, round-off errors can cause one or more eigenvalues of P to become zero or even negative. This is particularly likely if an eigenvalue is theoretically very close to zero; numerical noise can then push its computed value into the negative domain.3 A non-positive definite P matrix is theoretically impossible for a covariance matrix and can lead to nonsensical results, such as negative computed variances or an ill-conditioned Kalman gain calculation, ultimately causing the filter to diverge.
The standard covariance update equation, Pk∣k​=(I−Kk​Hk​)Pk∣k−1​, is known to be particularly susceptible to these numerical problems, especially the loss of positive definiteness, because it involves a subtraction that can lead to catastrophic cancellation if Pk∣k−1​ and Kk​Hk​Pk∣k−1​ are large and nearly equal.3 If P loses positive definiteness, the filter weighting coefficients (Kalman gain) computed using this matrix become erroneous, leading to incorrect state updates.7
The degradation of the P matrix's properties can sometimes be a "silent failure" in its initial stages. The filter might continue to produce state estimates that appear plausible for a period, even as the underlying covariance calculations are becoming corrupted. The Kalman gain calculations will still execute, but the resulting gains will be mathematically unsound. Initially, the impact of these corrupted gains on the final state estimate might be small and masked by measurement noise or other system uncertainties. However, as the corruption of P worsens with each iteration, the Kalman gains become increasingly erroneous, leading to progressively worse state estimates and eventual, noticeable divergence. This insidious, gradual decay means that diagnosing the onset of numerical divergence can be challenging unless the properties of P are explicitly monitored at each step.
Table 1: Principal Causes of Kalman Filter Divergence

Cause Category	Specific Cause	Description	Key Manifestations/Contributing Factors	Primary Impact on Filter	
Model Inaccuracies	System Dynamics Misrepresentation	Errors in the state transition model (F) or control input model (B).	Incorrect F matrix, unmodeled forces, dynamical biases (e.g., constant unmodeled acceleration, incorrect drag coefficient).7	Systematic prediction errors, causing estimate to drift from true state.	
	Erroneous Noise Covariance Specification (Q and R)	Incorrect statistical characterization of process noise (Q) or measurement noise (R).	Q too small ("filter smugness," overconfidence in model), Q too large (noisy estimates); R too small (over-trusts noisy data), R too large (ignores measurements).6	Over/under-weighting of measurements vs. model predictions; filter becomes unresponsive or overly sensitive to noise.	
	Unaccounted Dynamics and State Vector Omissions	Failure to include all relevant dynamic variables in the state vector x.	Omission of state variables (deliberate for complexity reduction or inadvertent).4	Inability of the filter to accurately model or predict system behavior, leading to persistent errors.	
	Incorrect Initial Conditions (x0​,P0​)	Grossly inaccurate initial state estimate (x0​) or initial error covariance (P0​).	P0​ too small (overly confident in a wrong x0​), x0​ far from true state.10	Slow convergence, prolonged transient errors, or failure to converge if compounded by other model errors.	
Numerical Instability	Finite Precision Arithmetic and Round-Off Error Accumulation	Errors introduced by implementing filter equations on digital computers with finite word length.	Accumulation of small round-off errors over many iterations, especially with single precision or long run times.3	Gradual degradation of state estimate and covariance accuracy, eventually leading to large errors.	
	Loss of Covariance Matrix Properties (Symmetry, Positive Definiteness)	The computed state error covariance matrix P loses its theoretical properties due to numerical errors.	P becomes non-symmetric or non-positive definite (e.g., negative eigenvalues/variances).3	Unreliable/nonsensical Kalman gains, unstable filter behavior, and eventual divergence.	

III. Strategies for Preventing and Mitigating Divergence
Given the potentially severe consequences of Kalman filter divergence, a variety of techniques have been developed to enhance its robustness and reliability. These strategies can be broadly categorized into methods that fortify the numerical integrity of the computations, those that improve adaptability to model uncertainties, paradigms that employ inherently robust filtering designs, and practical adjustments including regularization.
A. Fortifying Numerical Integrity
This class of solutions addresses divergence caused by the accumulation of numerical errors in finite-precision arithmetic, focusing primarily on maintaining the mathematical properties of the state error covariance matrix P.
1. The Joseph Form for Covariance Update
One of the most well-known methods to improve the numerical stability of the covariance update step is the use of the "Joseph form" or stabilized form of the equation. The standard update for the a posteriori error covariance Pk∣k​ can be numerically unstable. The Joseph form is given by:
Pk∣k​=(I−Kk​Hk​)Pk∣k−1​(I−Kk​Hk​)T+Kk​Rk​KkT​
where Pk∣k−1​ is the a priori error covariance, Kk​ is the Kalman gain, Hk​ is the observation matrix, and Rk​ is the measurement noise covariance matrix.11
While this formulation is computationally more expensive than the simpler, more common form (Pk∣k​=(I−Kk​Hk​)Pk∣k−1​), it offers significantly better numerical stability.11 It is less sensitive to round-off errors and is more robust in maintaining the crucial properties of symmetry and positive definiteness of the P matrix, especially in systems with many state variables (large degrees of freedom) or when implemented with low arithmetic precision.3 The Joseph form is also particularly recommended if a non-optimal Kalman gain is deliberately used, for instance, in certain adaptive or robust filtering schemes.9
The enhanced stability of the Joseph form arises from its algebraic structure. The term (I−Kk​Hk​)Pk∣k−1​(I−Kk​Hk​)T is a quadratic form (APAT). If Pk∣k−1​ is symmetric and positive semi-definite, this term will also be symmetric and positive semi-definite. Similarly, the term Kk​Rk​KkT​ will be symmetric and positive semi-definite if Rk​ (which is a covariance matrix) possesses these properties. The sum of two symmetric, positive semi-definite matrices is also symmetric and positive semi-definite. This structure inherently promotes the preservation of these essential mathematical properties, making it more resilient to numerical errors that might otherwise cause Pk∣k​ to become non-symmetric or lose positive definiteness, which are common precursors to divergence.
2. Square-Root Kalman Filtering (SRKF) Techniques
Square-Root Kalman Filtering (SRKF) encompasses a family of algorithms that propagate a square root (e.g., a Cholesky factor Sk​ where Pk​=Sk​SkT​) of the state error covariance matrix Pk​, rather than propagating Pk​ itself.12 This approach offers superior numerical stability for several reasons:
●	Reduced Dynamic Range: The condition number of a matrix square root Sk​ is generally much smaller than that of Pk​ (approximately the square root). Propagating quantities with a smaller dynamic range reduces their susceptibility to round-off errors during intermediate calculations.
●	Enforced Positive Definiteness: By working with the square-root factors, the positive definiteness (or semi-definiteness) of the covariance matrix is inherently maintained. When the full covariance matrix Pk​ is needed, it is reconstructed from its factor (e.g., Pk​=Sk​SkT​), which mathematically guarantees its symmetry and positive semi-definiteness, provided Sk​ is real.
SRKF methods effectively manage numerical health proactively by operating on quantities (the matrix square roots) that are inherently better conditioned than the full covariance matrix. This can prevent numerical issues from arising in Pk​ in the first place, rather than merely mitigating them after Pk​ has already begun to degrade. Various SRKF algorithms exist, differing in the type of factorization used (e.g., Cholesky, U-D decomposition) and the specifics of the propagation equations. Historically, SRKF techniques were crucial for implementations on computationally constrained platforms with limited word lengths, such as the Apollo flight computer.13 While these methods generally involve increased computational complexity compared to the standard Kalman filter equations operating on Pk​ directly, the gain in numerical precision and stability often justifies the additional cost in demanding applications.13
B. Enhancing Adaptability to Model Uncertainties
When divergence is primarily due to inaccuracies in the system model or noise statistics, adaptive techniques can be employed. These methods adjust the filter's parameters online based on the observed data, making the filter less sensitive to initial mis-specifications or changes in the operating environment.
1. Adaptive Kalman Filtering (AKF)
Adaptive Kalman Filtering (AKF) refers to a class of algorithms that dynamically adjust certain parameters of the Kalman filter—most commonly the process noise covariance Q, the measurement noise covariance R, or sometimes even elements of the system model matrices (F,H)—during its operation.14 The adaptation is typically based on an analysis of the filter's innovation sequence (also known as residuals), vk​=yk​−Hk​x^k∣k−1​, where yk​ is the current measurement and Hk​x^k∣k−1​ is the predicted measurement.
For an optimal Kalman filter, the innovation sequence should be a zero-mean, white (uncorrelated in time) Gaussian process with a known theoretical covariance, Sk​=Hk​Pk∣k−1​HkT​+Rk​.14 If the actual statistics of the innovation sequence deviate significantly from these theoretical properties (e.g., it becomes biased, correlated, or its covariance does not match Sk​), it indicates that the filter is suboptimal, likely due to model mismatch or incorrect noise assumptions. AKF algorithms use these deviations to adjust Q or R (or other parameters) in a direction that attempts to restore the desired properties of the innovations sequence, thereby improving filter performance and preventing divergence. Various techniques exist, such as covariance matching, where Q or R are adjusted so that the sample covariance of the innovations matches its theoretical value.14
AKF can be conceptualized as a higher-level feedback control system. In this analogy, the Kalman filter itself is the "plant" being controlled. The "output" of this plant is the innovation sequence. The adaptive mechanism acts as a "controller," monitoring the statistical properties of this output. If these properties deviate from their desired setpoints (e.g., whiteness, zero mean, theoretical covariance), the controller adjusts the "inputs" to the Kalman filter plant—namely, its parameters like Q and R. This feedback loop continuously strives to maintain the filter's optimality, or at least its stability and reasonable performance, even when faced with unknown or time-varying characteristics of the system or its environment.
2. Fading Memory and Exponential Weighting Approaches (e.g., Adaptive Fading Kalman Filter - AFKF)
Fading memory Kalman filters, including the Adaptive Fading Kalman Filter (AFKF), are designed to give more importance to recent measurements and less to older data.16 This is particularly useful when the system model is not perfectly known, may be time-varying, or when initial model errors have a lingering, detrimental effect on the filter's performance. The core idea is to prevent the filter from becoming too confident (i.e., its P matrix becoming too small) based on a long history of data that might have been processed using a stale or inaccurate model.
This is typically achieved by introducing a "forgetting factor" λ≥1 (sometimes denoted j(k)). This factor is used to artificially inflate the a priori state error covariance matrix before the measurement update, for example, by modifying the prediction step for P:
Pk∣k−1​=λk​Fk−1​Pk−1∣k−1​Fk−1T​+Qk−1​
An increased Pk∣k−1​ leads to a larger Kalman gain Kk​, which in turn gives more weight to the current measurement residual in the update step. This makes the filter more responsive to new information and less reliant on its past estimates, which might be tainted by model errors.17 The AFKF specifically aims to adaptively adjust this forgetting factor based on the filter's performance, often using the innovation sequence.16
Fading memory techniques essentially introduce a form of controlled "amnesia" into the filter. By systematically de-weighting or "forgetting" older information (achieved indirectly by inflating the predicted covariance, which increases reliance on new data), the filter is compelled to rely more on recent observations. This is beneficial when the underlying system model is uncertain or may be slowly changing over time. It prevents the filter from becoming entrenched in outdated assumptions derived from a long history processed with a potentially flawed model. This mechanism allows the filter to adapt more quickly to changes in system behavior or to discount the cumulative effects of persistent model errors that would otherwise lead to divergence.
C. Employing Robust Filtering Paradigms
Robust filtering approaches are designed from the ground up to be less sensitive to violations of the standard Kalman filter assumptions, particularly uncertainties in the system model.
1. Principles of Robust Kalman Filtering (Minimax, Risk-Sensitive)
Robust Kalman filtering techniques aim to provide guaranteed performance over a predefined set or class of possible model uncertainties, rather than striving for optimality under a single, precisely known nominal model.20 This often involves:
●	Minimax Strategies: These approaches seek to minimize the worst-case estimation error, assuming that the true system model could be any model within a specified uncertainty set. The filter is designed to perform optimally (or adequately) for the "least favorable" model perturbation within this set.20
●	Risk-Sensitive Criteria: These methods modify the optimization criterion to penalize large estimation errors more heavily than smaller ones. This leads to filters that are more cautious and less prone to large deviations when the true system differs from the nominal model.
The resulting robust filters often retain a Kalman-like recursive structure but with modified gain computations. For instance, some minimax approaches based on divergence measures (like Tau-divergence to define the uncertainty ball around the nominal model) yield filters where the gain update resembles that of a risk-sensitive filter, often incorporating a time-varying risk sensitivity parameter that adapts to the perceived level of model mismatch.20
Robust Kalman filtering, particularly through minimax strategies, embodies a "pessimistic" design philosophy. It operates under the assumption that the true system model might be the most challenging one within a defined range of uncertainties. The filter is then engineered to deliver acceptable performance even under this worst-case scenario. This approach inherently sacrifices the optimality that a standard Kalman filter might achieve if the nominal model were perfectly correct. However, in return, it gains a guarantee of a certain level of performance across a broader spectrum of potential model variations. This contrasts sharply with the standard Kalman filter, which, while optimal for its assumed model, can perform very poorly if the true system dynamics deviate even slightly from those nominal assumptions.
2. The Consider Kalman Filter (Schmidt-Kalman Filter)
The Schmidt-Kalman filter, also known as the "consider filter," is a specialized technique designed to account for the uncertainty associated with certain parameters (often called "consider parameters," "nuisance parameters," or biases) without actually including them in the state vector to be estimated.21 Instead, the filter's covariance calculations are modified to reflect the influence of these unestimated but uncertain parameters on the error in the estimated states. This leads to a more realistic state error covariance matrix P for the estimated states and, consequently, more robust Kalman gains.
The primary benefits of the Schmidt-Kalman filter are:
●	Reduced Computational Complexity: By not augmenting the state vector with the "consider" parameters, the dimensionality of the main filter remains smaller, reducing computational load compared to estimating all uncertain parameters.22
●	Handling of Unobservable or Poorly Observable Biases: It is particularly useful for dealing with parameters that are unobservable (their effect cannot be uniquely determined from the available measurements) or poorly observable. Attempting to estimate such parameters can lead to ill-conditioning and filter instability. The Schmidt-Kalman filter provides a robust way to acknowledge their uncertainty without attempting direct estimation.21
●	Improved Covariance Realism: It ensures that the covariance matrix P more accurately represents the true distribution of errors in the estimated states by accounting for the contribution of the "consider" parameter uncertainties.22
The Consider Kalman Filter offers a pragmatic compromise in situations where certain sources of error or uncertainty are known to exist but are difficult or undesirable to estimate directly. Fully augmenting the state vector to estimate every uncertain parameter can be computationally prohibitive, increase the risk of observability problems, and complicate filter tuning. Conversely, completely ignoring these parameters leads to an incomplete model, an optimistically small (and therefore incorrect) covariance matrix P for the primary states, and a high risk of divergence. The Schmidt-Kalman filter navigates a middle path: it keeps the dimension of the estimated state vector manageable by not estimating the "consider" parameters, but it explicitly incorporates their statistical uncertainty into the calculation of the covariance matrix P for the actively estimated states and, through P, into the Kalman gain. This makes the primary state estimation process "aware" of these additional unmodeled error sources, resulting in a more realistically inflated P matrix and consequently more cautious (and robust) Kalman gains. It is a practical approach to improve robustness without incurring the full cost and complexity of estimating every conceivable uncertainty.
D. Regularization and Practical Adjustments
These are often simpler, sometimes heuristic, methods employed to enhance filter stability and prevent common issues.
1. Covariance Regularization (Inflation, Additive Adjustments)
Covariance regularization techniques involve modifying the covariance matrices (P, Q, or R) to improve filter stability and prevent overconfidence.
●	Covariance Inflation: This involves artificially inflating the state error covariance matrix P (usually the a priori Pk∣k−1​) or the process noise covariance Q by multiplying it by a scalar factor α>1. This prevents the filter from becoming too confident in its estimates (i.e., P becoming too small), ensuring it remains responsive to new measurements. It is a common technique, especially in Ensemble Kalman Filters (EnKF), to account for unmodeled errors or sampling errors.9
●	Additive Adjustments: Adding small positive values to the diagonal elements of Q or R (e.g., Qreg​=Q+ϵI, where ϵ is a small positive scalar and I is the identity matrix) can help maintain their positive definiteness and prevent numerical singularities that might arise if these matrices become ill-conditioned or rank-deficient.9
Covariance regularization techniques can be viewed as an "insurance policy" against "unknown unknowns"—sources of error, model mismatch, or numerical inaccuracies that are not explicitly accounted for in the filter design. By forcing the filter to maintain a certain minimum level of uncertainty (i.e., preventing P from becoming unrealistically small or Q/R from being singular), these methods ensure that the filter remains somewhat responsive to new measurements and retains a degree of adaptability. Even if the known components of the model suggest very high confidence, regularization introduces a baseline level of caution, acknowledging that the model is unlikely to be perfect and that unexpected deviations or unmodeled effects might occur.
2. Judicious Filter Tuning and Resetting Strategies
"Filter tuning" is the process of adjusting filter parameters, primarily Q, R, and the initial covariance P0​, based on observed filter performance, often through trial-and-error or more systematic analysis of residuals. This is a common and often necessary practice to combat divergence, especially when model errors are suspected.3
In more extreme cases, if divergence is detected or strongly suspected, and other methods are insufficient or too complex to implement quickly, a "filter reset" might be employed. This involves re-initializing the state estimate x^ and/or the error covariance matrix P to their initial values (x^0​,P0​) or to some state of high uncertainty.5 Periodic restarting of a time-varying Kalman filter can be an effective cure for problems where certain system modes are undisturbed by the modeled process noise and are neutrally stable, as these can lead to an unwarranted decrease in the corresponding elements of P and subsequent divergence.5
Filter resetting is a rather drastic measure, akin to performing a "hard reset" on the estimation process. It is typically used when the filter's internal state (its estimate x^ and covariance P) is deemed so corrupted by accumulated errors or numerical degradation that incremental corrections via the normal update mechanism are unlikely to restore satisfactory performance. A reset acknowledges that the filter has fundamentally "lost track" of the true state and effectively discards its accumulated (and now erroneous) history to start afresh. The hope is that, upon restarting, perhaps with better initial conditions or after a transient system anomaly has passed, the filter will be able to reconverge. This action essentially forces the filter to rely heavily on new measurements, much like during its initial startup phase, because the re-initialized P (if set to a large uncertainty) will yield large Kalman gains.
IV. Comparative Insights and Practical Guidance
Addressing Kalman filter divergence requires a nuanced understanding of the available solutions, as each comes with its own set of mechanisms, benefits, limitations, and computational implications. Selecting the most appropriate technique depends heavily on the specific nature of the divergence, the characteristics of the system, and the operational constraints.
A. A Comparative Overview of Divergence Solutions
No single solution universally triumphs in mitigating Kalman filter divergence; a "no free lunch" principle applies. Numerical stability methods like the Joseph form and Square-Root Kalman Filtering (SRKF) offer enhanced robustness against round-off errors but typically at the cost of increased computational load.11 While the Joseph form provides a more robust update equation for the covariance matrix P itself 11, SRKF proactively manages numerical health by operating on matrix square roots which are inherently better conditioned.12
Adaptive methods, such as Adaptive Kalman Filtering (AKF) and Fading Memory filters, provide mechanisms for the filter to learn from data and adjust its parameters (like Q and R) online, making them valuable when model parameters are uncertain or time-varying.14 However, these adaptive schemes can introduce their own complexities in terms of design, tuning of adaptation mechanisms, and convergence guarantees, and they may not always achieve true optimality.14
Robust filtering paradigms, including minimax approaches and the Consider Kalman Filter, are designed to provide guaranteed performance bounds over a class of uncertainties.20 Minimax filters often sacrifice nominal performance for worst-case robustness 20, while the Consider Filter offers a compromise by accounting for unestimated parameter uncertainties without the full computational burden of estimating them.22
Simpler, often heuristic, methods like covariance regularization (e.g., inflation) and filter resetting serve as practical tools. Regularization can prevent overconfidence and maintain matrix properties 9, while resetting offers a last resort when the filter state is deemed unrecoverable.5 These methods are often easier to implement but may lack the theoretical rigor or guaranteed performance of more sophisticated techniques. The choice of solution is therefore highly application-dependent, requiring a careful balance of factors such as the suspected cause of divergence, available computational resources, the nature and extent of model uncertainties, and the required level of estimation accuracy and reliability.
Table 2: Comparative Summary of Divergence Mitigation Techniques

Technique	Core Mechanism	Primary Benefits	Key Limitations/Trade-offs	Typical Use Cases/Divergence Types Addressed	Key Snippets	
Joseph Form (Covariance Update)	Uses a numerically stabilized equation to update $P_{k\$	k}.	Improved numerical stability; maintains symmetry and positive definiteness of P.	Higher computational cost than standard form.	Finite precision errors, especially in high-DOF systems or with low precision arithmetic.	
Square-Root Kalman Filtering (SRKF)	Propagates square-root factors of P instead of P itself.	Superior numerical stability; reduced dynamic range of propagated quantities; inherently maintains positive definiteness of P.	Increased computational complexity; more complex implementation.	Severe numerical stability challenges; limited word-length processors; high dynamic range systems.	12	
Adaptive Kalman Filtering (AKF)	Online adjustment of filter parameters (e.g., Q,R) based on innovation sequence statistics.	Adapts to unknown or time-varying noise characteristics or model parameters; can improve accuracy in non-ideal conditions.	Increased complexity; tuning of adaptation mechanism; convergence not always guaranteed; potential for instability if adaptation is too aggressive.	Poorly known or changing Q/R; some types of model mismatch.	14	
Fading Memory / Exponential Weighting (e.g., AFKF)	Gives more weight to recent data by inflating $P_{k\$	k-1} via a forgetting factor.	Tracks time-varying systems or parameters; reduces influence of old, potentially incorrect model assumptions.	Heuristic choice of forgetting factor (unless adaptive); can degrade performance if system is truly time-invariant and well-modeled.	Model parameter drift; unmodeled slow changes in system dynamics; initial model errors.	
Robust Kalman Filtering (Minimax, Risk-Sensitive)	Designs filter for worst-case performance over a set of model uncertainties.	Guaranteed performance bounds over a class of models; less sensitive to specific model perturbations.	May be conservative (suboptimal for nominal model); mathematically complex to derive and implement.	Significant, bounded model uncertainty; applications requiring high reliability under various conditions.	20	
Consider Kalman Filter (Schmidt-Kalman)	Accounts for uncertainty of "consider" parameters in P without estimating them in the state vector.	Reduced computational load vs. full state augmentation; handles unobservable/poorly observable biases; more realistic covariance for estimated states.	Suboptimal compared to full state estimation if "consider" parameters were perfectly observable and estimated; requires knowledge of "consider" parameter statistics.	Unmodeled but known biases (e.g., sensor biases); parameters with poor observability.	21	
Covariance Regularization (Inflation, Additive Noise)	Artificially inflates P or Q, or adds small values to diagonal of Q/R.	Prevents overconfidence; maintains positive definiteness of Q/R; simple to implement.	Heuristic; may lead to suboptimal performance if not tuned carefully; can mask underlying problems.	General robustness enhancement; preventing P from becoming too small; ensuring Q/R are well-conditioned.	9	
Filter Resetting	Re-initializes state estimate and/or covariance matrix.	Simple way to recover from severe divergence; discards corrupted filter history.	Loss of all past information; can cause transients in estimates; addresses symptom, not root cause.	Catastrophic filter failure; detected gross divergence; intermittent severe disturbances.	5	

B. Selecting Appropriate Mitigation Techniques
Diagnosing the specific cause of divergence is the first crucial step in selecting an appropriate mitigation strategy. If numerical errors are suspected, particularly in applications involving high-dimensional state vectors, long operational durations, or limited computational precision, then techniques like Square-Root Kalman Filtering or employing the Joseph form for the covariance update should be primary considerations. SRKF offers a more fundamental solution by working with better-conditioned matrix factors, while the Joseph form provides a more robust update equation for the full covariance matrix.
When the primary concern is uncertainty or time-variation in the noise covariance matrices (Q and R), Adaptive Kalman Filtering techniques are strong candidates. These methods can estimate or adjust Q and R online, allowing the filter to adapt to the actual noise environment. Fading memory filters are also beneficial in such scenarios, or when the system model itself might be slowly changing, as they reduce the filter's reliance on outdated information.
If significant, known but unmodeled dynamics exist, or if there is substantial uncertainty about specific parameters that are difficult to estimate (e.g., sensor biases), then robust filtering approaches like the Consider Kalman Filter (Schmidt-Kalman filter) may be suitable. The Consider Filter allows these uncertainties to be accounted for in the covariance calculations without increasing the state vector size. More general robust Kalman filtering methods based on minimax or risk-sensitive principles can be employed when aiming for guaranteed performance across a broader class of model uncertainties, though these often come with higher analytical and computational complexity.
Covariance regularization techniques, such as inflation or adding small positive values to Q or R, are often useful as first-line-of-defense measures or as complementary additions to other methods. They are relatively simple to implement and can help prevent filter overconfidence and maintain numerical stability. Filter resetting is generally a measure of last resort, used when divergence is severe and other methods are either ineffective or too slow to recover.
A key diagnostic tool for assessing filter health and identifying potential divergence, often before it becomes catastrophic, is the innovation sequence (vk​=yk​−Hk​x^k∣k−1​). In an optimally performing Kalman filter, the innovation sequence should be a zero-mean, white noise process with a predictable covariance.14 Deviations from these properties—such as a non-zero mean (bias), temporal correlation, or a sample covariance that does not match the theoretical Hk​Pk∣k−1​HkT​+Rk​—are strong indicators of model mismatch, incorrect noise assumptions, or other problems that could lead to divergence.9 Monitoring the statistical properties of the innovation sequence can therefore provide valuable early warnings and insights into the nature of the problem, guiding the selection of appropriate corrective actions or adjustments to the filter design or tuning parameters.
V. Conclusion: Ensuring Reliable Kalman Filter Performance
Kalman filter divergence remains a significant challenge in the practical application of this otherwise powerful estimation tool. It stems fundamentally from the inevitable imperfections in mathematical models of real-world systems and the inherent limitations of numerical computation. The consequences of divergence can range from degraded estimation accuracy to complete failure of control and navigation systems, underscoring the critical importance of addressing this issue.
A diverse array of strategies has been developed to combat divergence, each with its own mechanisms, advantages, and drawbacks. These range from algorithmic modifications that enhance numerical stability (such as the Joseph form and Square-Root Kalman Filtering), to adaptive techniques that allow the filter to learn and adjust to model uncertainties online (like Adaptive Kalman Filtering and fading memory approaches), to robust filter designs (including minimax strategies and the Consider Kalman Filter) that aim for guaranteed performance over a class of uncertainties, and finally, to practical regularization methods and resetting protocols.
Successfully avoiding or mitigating Kalman filter divergence is not merely about applying a specific algorithmic fix when problems manifest. It requires a more holistic approach that begins with proactive design choices. This includes careful system modeling, thoughtful selection of the state vector, realistic characterization of noise sources, and consideration of the numerical demands of the application. For instance, opting for a Square-Root Kalman Filter from the outset in applications known to be numerically challenging, or designing a robust filter when significant model uncertainty is anticipated, can preemptively address potential divergence issues.
Furthermore, ensuring reliable filter performance necessitates continuous vigilance throughout the operational life of the filter. This involves ongoing monitoring of the filter's behavior, with particular attention to the statistical properties of the innovation sequence, which serves as a crucial real-time indicator of filter health. Periodic re-evaluation of model assumptions and noise characteristics may also be necessary, especially in systems that operate in changing environments or over extended periods.
Ultimately, the selection and implementation of effective divergence mitigation strategies demand a thorough understanding of the specific system under consideration, a clear identification of potential error sources, and a realistic assessment of computational constraints and performance requirements. By combining careful design, appropriate algorithmic choices, and diligent monitoring, engineers and researchers can significantly enhance the robustness and reliability of Kalman filter applications, harnessing their full potential in a wide range of complex estimation problems.
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