Ch. 4: (Repeat the simulations of Examples 4.4 with a continuous time system mod
ID: 2249431 • Letter: C
Question
Ch. 4: (Repeat the simulations of Examples 4.4 with a continuous time system model and a discrete Kalman filter)
Do not generate and save the measurements first, then use them with the Kalman filter.
4.4 The system shown is driven by two independent Gaussian white sources u1(t) and u2(t). Their spectral functions are given by Sal = 4(ft/s)2/Hz 2 16(ft/s)2/Hz Let the state variables be chosen as shown on (he diagram, and assume that noisy measurements of xi, are obtained at unit intervals of time. A discrete Kalman filter model is desired. Find k and Qk for this model u2(t) 2 uj(t) Figure P4.4Explanation / Answer
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 using Bayesian inference and 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.
The Kalman filter has numerous applications in technology. A common application is for guidance, navigation, and control of vehicles, particularly aircraft and spacecraft.[1] Furthermore, the Kalman filter is a widely applied concept in time series analysis used in fields such as signal processing and econometrics. Kalman filters also are one of the main topics in the field of robotic motion planning and control, and they are sometimes included in trajectory optimization. The Kalman filter also works for modeling the central nervous system's control of movement. Due to the time delay between issuing motor commands and receiving sensory feedback, usage of the Kalman filter supports the realistic model for making estimates of the current state of the motor system and issuing updated commands.[2]
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.
The Kalman filter does not make any assumption that the errors are Gaussian.[3] However, the filter yields the exact conditional probability estimate in the special case that all errors are Gaussian-distributed.
Extensions and generalizations to the method have also been developed, such as the extended Kalman filter and the unscented Kalman filter which work on nonlinear systems. The underlying model is a Bayesian model similar to a hidden Markov model except that the state space of the latent variables is continuous and all latent and observed variables have Gaussian distributions.