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Kalman Filter Paper Planning

Dec 11, 20246 min read

Reference Used

10/10

Abstract

  1. Introduction: A brief statement of the problem or issue 
  2. Purpose: A description of the purpose of the work 
  3. Method: A description of the research method and design 
  4. Result: A summary of the major findings 
  5. Conclusion: A description of the conclusions reached

Background of the Study

  1. History
    1. It is a system over 20 years old that has applications not only in engineering but also in economic systems (e.g., applications in the car, military, and biomedicine industries).
    2. Although potential measurement errors can exist, the Kalman Filter was developed by Rudolf E. Kalman in 1960 to predict non-observable state variables using observable ones.
    3. The studies conducted by the Dynamic Analysis Branch of NASA, which was headed by Dr. Schmidt from the Ames Research Center (ARC), led to what is currently referred to as the Extended Kalman Filter (EKF).
    4. In brief, the studies were able to accomplish the following:
      1. Apply the original Kalman theory to nonlinear problems
      2. The emergence of EKF, a method that minimizes problems found in nonlinear systems after a linearization is applied over the best real state estimation
      3. The break down and reorganization of the Kalman algorithm to process measurements at any arbitrary time interval.
      4. The demonstration of what is possible with the Kalman filter through its utilization in solving nonlinear orientation and navigation problems occurring in spaceship simulations.
      5. The spread of the results of the demonstration to promote its possible usage in control and navigation systems of the Apollo spacecraft.
      6. The promulgation of the Kalman filter in scientific and aerospatial units.
  2. Controversies/Problems
    1. One study leveraged the Kalman filter for iterative learning control (ILC)—an open loop approach utilized in eliminating tracking errors arising from control systems that keep performing the same tracking maneuver.
    2. They used real world data over a mathematical model that attempts to yield zero tracking error to ensure more consistent results when practically applied.
    3. In analyzing the Kalman Filter’s application in repetitive control (RC), it was found that it performed worse with an imperfect model. That is why, one must carefully consider whether the tradeoffs involving the reduced random variations is worth it considering the introduction of deterministic error.
    4. The same situation occurs in ILC when running in time steps during each iteration.
  3. Why Kalman
    1. The Kalman filter is a prevalent algorithm because of its low computational requirements, well design recursive properties, and its representation of the optimal estimator for one dimensional linear systems that assume Gaussian error statistics, and its appropriateness in real-time implementation.
    2. They are helpful for modeling dynamic systems and estimation because they connect real-time measurements and predict the state of system parameters through time intervals.

Significance of the Study

  1. Purpose of State Estimation
    • A way to find the best estimate for a state process
    • Uses measurements that reflect parts of the process state
    • Ultimate goal is to provide a representation that closely mirrors the process state
    • They can accommodate for existence of inconsistencies or differences found within models and actual measurements.
  2. Importance of Noise Minimization
    1. High sensitivity sensors are becoming increasingly ubiquitous for enhancing system performance.
    2. Sensors are vulnerable to noise and external disturbances when capturing environmental data
    3. Noise can negatively affect the performance of a controller by distorting the real measured data.
    4. This can result in hampering the control system in reaching its potential, or, at worst, prevent it from controlling the system.
  3. Navigation Systems
    1. In Inertial Navigation Systems (INS) and Global Navigation Satellite Systems (GNSS), Kalman Filters are often used for approximating navigation parameters like aircraft position, velocity, and attitude angle. Nonetheless, some measurement noise are present; consequently, they need to be corrected through external measurement information provided by the GNSS signals.
    2. The Kalman filter obtains the optimal state variable estimations by modifying the optimal gain through the measurement noise covariance and the state ratio.
  4. Purpose of Kalman Filter
    1. It is an optimal state estimator designed with linearity and Gaussian noise in mind.
      1. Optimal state estimation means best possible way of estimating the state of a general system.
    2. It is effective because it considers the fact that sensors have some amount of errors when compared to the actual state.
    3. As mentioned previously, the approach it takes involves delving into the variability of multiple unobserved states by taking advantage of past observed states.
  1. Kalman Filter and PID Controller 1
    1. The Kalman filter algorithm is used by the study to solve the data-driven PID parameters tuning problem
    2. PID parameters tuning is important due to its influence on control performance of closed-loop systems.
    3. In the industrial world, there is recent rise of fictitious reference iterative tuning (FRIT) and virtual reference feedback tuning (VRFT) representations in data-driven controller design schemes for PID controllers.
    4. They chose Kalman filter due to its ability to deliver good estimation results even with noisy environments and nonlinear systems.
  2. Kalman Filter, PID Controller, and DC Motor
    1. Even with the variety of tuning models created to ensure optimal PID control performance, the lack of consideration on the effect of noise means that its performance will not yield optimal results in non-ideal conditions.
    2. In this regard, this study aims to create a PID control system that mitigates the impact of noise through the Kalman filter.
    3. In particular, the Kalman filter was utilized for minimizing the noise that was tied to the input signal so that the actual output resembles the expected output much more closely.
    4. The results of their study suggests that the Kalman filter improved the performance of the PID control system when compared to the ordinary version.
    5. To elaborate further, by comparing a PID control system with a Kalman filter implemented and another PID without the Kalman filter, the study reported that the motor speed response from their experiment with the Kalman-filter based PID resulted in a lower integral of absolute error (IAE), and also lower overshoot than the PID without the Kalman-filter.
  3. Kalman Filter Improvements
    1. High precision sensors are generally expected for intelligent terminals, thereby stressing the need for effective noise filters. In other words, preventing noise from generating disturbance in the system is crucial especially since the accuracy of measurements have direct effects on the control system.
    2. Paper improved the traditional Kalman filter by introducing neuro units. Their solution involves the sequential and separate application of the Kalman filter and artificial neural network modeling (ANN).
    3. To be exact, the neural network processes the data both before and after the filtering process, primarily to make the noise filter extract better outputs.
    4. This produced a more optimized filtering process for mitigating the problems arising from an imperfect system model and impractical hypothetical parameters.
    5. Simulation and tests indicate that the new design was an improvement in terms of its noise elimination performance within the soft computing solution.

Analysis of Data

  • noise covariance Q and measurement error covariance R for evaluating the performance of algorithm
  • step response utilized for behavior: underdamped, critically damped, and overdamped

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