The Kalman filter is the minimum-variance state estimator for linear dynamic systems with Gaussian noise. Even if the noise is non-Gaussian, the Kalman filter is the best linear estimator. For nonlinear systems it is not possible, in general, to derive the optimal state estimator in closed form, but various modifications of the Kalman filter can be used to estimate the state. These modifications include the extended Kalman filter, the unscented Kalman filter, and the particle filter. Although the Kalman filter and its modifications are powerful tools for state estimation, we might have information about a system that the Kalman filter does not incorporate. For example, we may know that the states satisfy equality or inequality constraints. In this case we can modify the Kalman filter to exploit this additional information and get better filtering performance than the Kalman filter provides. This paper provides an overview of various ways to incorporate state constraints in the Kalman filter and its nonlinear modifications. If both the system and state constraints are linear, then all of these different approaches result in the same state estimate, which is the optimal constrained linear state estimate. If either the system or constraints are nonlinear, then constrained filtering is, in general, not optimal, and different approaches give different results.

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Kalman filtering with state constraints: a survey of linear and nonlinear algorithms

Importance In-hospital cardiac arrest is common and associated with a high mortality rate. Despite this, in-hospital cardiac arrest has received little attention compared with other high-risk cardiovascular conditions, such as stroke, myocardial infarction, and out-of-hospital cardiac arrest. Observations In-hospital cardiac arrest occurs in over 290 000 adults each year in the United States. Cohort data from the United States indicate that the mean age of patients with in-hospital cardiac arrest is 66 years, 58% are men, and the presenting rhythm is most often (81%) nonshockable (ie, asystole or pulseless electrical activity). The cause of the cardiac arrest is most often cardiac (50%-60%), followed by respiratory insufficiency (15%-40%). Efforts to prevent in-hospital cardiac arrest require both a system for identifying deteriorating patients and an appropriate interventional response (eg, rapid response teams). The key elements of treatment during cardiac arrest include chest compressions, ventilation, early defibrillation, when applicable, and immediate attention to potentially reversible causes, such as hyperkalemia or hypoxia. There is limited evidence to support more advanced treatments. Post–cardiac arrest care is focused on identification and treatment of the underlying cause, hemodynamic and respiratory support, and potentially employing neuroprotective strategies (eg, targeted temperature management). Although multiple individual factors are associated with outcomes (eg, age, initial rhythm, duration of the cardiac arrest), a multifaceted approach considering both potential for neurological recovery and ongoing multiorgan failure is warranted for prognostication and clinical decision-making in the post–cardiac arrest period. Withdrawal of care in the absence of definite prognostic signs both during and after cardiac arrest should be avoided. Hospitals are encouraged to participate in national quality-improvement initiatives. Conclusions and Relevance An estimated 290 000 in-hospital cardiac arrests occur each year in the United States. However, there is limited evidence to support clinical decision making. An increased awareness with regard to optimizing clinical care and new research might improve outcomes.

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In-Hospital Cardiac Arrest: A Review

Adult Advanced Life Support: 2020 International Consensus on Cardiopulmonary Resuscitation and Emergency Cardiovascular Care Science With Treatment Recommendations

Constrained Bayesian state estimation – A comparative study and a new particle filter based approach

Advanced life support: 2010 International Consensus on Cardiopulmonary Resuscitation and Emergency Cardiovascular Care Science With Treatment Recommendations.

Both constrained and unconstrained optimization problems regularly appear in recursive tracking problems engineers currently address – however, constraints are rarely exploited for these applications. We define the Kalman Filter and discuss two different approaches to incorporating constraints. Each of these approaches are first applied to equality constraints and then extended to inequality constraints. We discuss methods for dealing with nonlinear constraints and for constraining the state prediction. Finally, some experiments are provided to indicate the usefulness of such methods.

/pdf/kalman-filtering-with-equality-and-inequality-state-14mhnrjsyp.pdf

Kalman Filtering with Equality and Inequality State Constraints

A Lymphedema Surveillance Program for Breast Cancer Patients Reveals the Promise of Surgical Prevention.

We discuss two separate techniques for Kalman Filtering in the presence of state space equality constraints. We then prove that despite the lack of similarity in their formulations, under certain conditions, the two methods result in mathematically equivalent constrained estimate structures. We conclude that the potential benefits of using equality constraints in Kalman Filtering often outweigh the computational costs, and as such, equality constraints, when present, should be enforced by way of one of these two methods.

Kalman Filtering in the Presence of State Space Equality Constraints

Variability in Interpretation of Cardiac Standstill Among Physician Sonographers

We discuss two separate techniques for Kalman filtering in the presence of state space equality constraints. We discuss these methods and then prove that despite the lack of similarity in their formulations, under certain conditions, the two methods result in mathematically equivalent constrained estimate structures. We then discuss numerical stability for implementations. We conclude that the potential benefits of using equality constraints in Kalman filtering often outweigh the computational costs, and as such, equality constraints, when present, should be enforced by way of one of these two methods.

/pdf/kalman-filtering-in-the-presence-of-state-space-equality-56zbxr2jrr.pdf

Nachi Gupta

Papers

Kalman Filtering with Equality and Inequality State Constraints

A Lymphedema Surveillance Program for Breast Cancer Patients Reveals the Promise of Surgical Prevention.

Kalman Filtering in the Presence of State Space Equality Constraints

Variability in Interpretation of Cardiac Standstill Among Physician Sonographers

Kalman Filtering in the Presence of State Space Equality Constraints