Linear Quadratic Estimation and Applications of Kalman Filter

1. Linear Quadratic Estimation using Kalman
Filter and Application
A Seminar Associated with
School of Computer Engineering, Kalinga Institute of Industrial Technology, DU
Saptarshi Mazumdar
1729058
Under the guidance of
Prof. Sujata Swain

2. C O N T E N T S
Part 01 Part 02 Part 03 Part 04
Introduction State Estimator Workflow Example Use

3. 01
Introduction to Kalman Filtering
In statistics and control theory, Kalman filtering, also known as linear quadratic estimation (LQE), is an algorithm that uses a
series of measurements observed over time, containing statistical noise and other inaccuracies, and produces estimates of
unknown variables that tend to be more accurate than those based on a single measurement alone, by estimating a joint
probability distribution over the variables for each timeframe. The filter is named after Rudolf E. Kálmán, one of the primary
developers of its theory.

4. 01 Introduction to Kalman Filtering
The algorithm works in a two-step process. In the prediction step, the Kalman filter produces estimates of the current state
variables, along with their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some amount of
error, including random noise) is observed, these estimates are updated using a weighted average, with more weight being given to
estimates with higher certainty. The algorithm is recursive. It can run in real time, using only the present input measurements and
the previously calculated state and its uncertainty matrix; no additional past information is required.
Why to Use Linear Quadratic Estimation

5. 02
State Estimation Algorithm
The Kalman filter keeps track of the estimated state of the system and the variance or
uncertainty of the estimate. The estimate is updated using a state transition model
and measurements.

6. 02 State Estimation Algorithm
Work Principle
The algorithm works in a two-step process. In the prediction step, the
Kalman filter produces estimates of the current state variables, along
with their uncertainties. Once the outcome of the next measurement
(necessarily corrupted with some amount of error, including random
noise) is observed, these estimates are updated using a weighted
average, with more weight being given to estimates with higher
certainty. The algorithm is recursive. It can run in real time, using only
the present input measurements and the previously calculated state
and its uncertainty matrix; no additional past information is required.

7. 01 State Estimation Algorithm

8. 03
Workflow of Kalman Filter
Noisy sensor data, approximations in the equations that describe the system evolution, and external factors that are not
accounted for all place limits on how well it is possible to determine the system's state. The Kalman filter deals effectively
with the uncertainty due to noisy sensor data and, to some extent, with random external factors. The Kalman filter
produces an estimate of the state of the system as an average of the system's predicted state and of the new
measurement using a weighted average. The relative certainty of the measurements and current state estimate is an
important consideration, and it is common to discuss the response of the filter in terms of the Kalman filter's gain. When
performing the actual calculations for the filter, the state estimate and covariances are coded into matrices to handle the
multiple dimensions involved in a single set of calculations

9. 03 Workflow
Fk: the state-transition model
Hk: the observation model;
Qk: the covariance of the process
noise
Rk: the covariance of the
observation noise
Bk: the control-input model, for
each time-step, k, as described
below.
The Kalman filter model
assumes the true state at time
k is evolved from the state at
(k − 1) according to

10. 01 Workflow
It is an efficient recursive filter that estimates the internal state of a linear dynamic system
from a series of noisy measurements. It is used in a wide range of engineering and
econometric applications from radar and computer vision to estimation of structural
macroeconomic models, and is an important topic in control theory and control systems
engineering. Together with the linear-quadratic regulator (LQR), the Kalman filter solves the
linear–quadratic–Gaussian control problem (LQG). The Kalman filter, the linear-quadratic
regulator, and the linear–quadratic–Gaussian controller are solutions to what arguably are the
most fundamental problems in control theory.

11. 04
Example Use

12. 04 Example Uses
GPS Monitoring Systems
US Navy Submarines
Aircraft
Computer Vision
Spacecraft
Recursive Elemination of
Gaussian Noise

13. THANK YOU
https://bit.ly/saptarshimazumdar