On Kalman Filtering. The 1960s: A Decade to Remember

Transcription

1 On Kalman Filtering A study of A New Approach to Linear Filtering and Prediction Problems by R. E. Kalman Mehul Motani February, 000 The 960s: A Decade to Remember Rudolf E. Kalman in 960 Research Institute for Advanced Studies (Baltimore) The Discrete-time Kalman Filter With Richard Bucy in 96 Bucy was with Johns Hopins Applied Physics Lab The Continuous-time Kalman Filter Kalman Filtering used widely in Control Systems, Signal Processing, Mehul Motani, 000

2 Some Mehul Motani, The Static Case First Measurement First Estimate ˆ ˆ = = Conditional PDF of position based on measurement ~Normal(, Mehul Motani, 000 4

3 The Static Case (cont.) Second Measurement, t t Second Estimate Conditional PDF of position based on measurement ~Normal(, ) ˆ ˆ =?? =?? How do we optimally combine the two Mehul Motani, The Combined Estimate Conditional PDF of position based on both measurements ~Normal( µ, ˆ ) Mehul Motani,

4 The Optimum Estimate is µ = ˆ σ = + ˆ = Mehul Motani, What does optimum mean? Unbiased (since it the conditional mean) Maimum Lielihood Estimate Least Squares Estimate Minimum Variance, Unbiased Estimate The Kalman filter is all of the Mehul Motani,

5 ˆ Another Loo The Predictor-Corrector Structure ˆ [ ] = + K ˆ ˆ ˆ = K ˆ Predict Mehul Motani, The Dynamic Case System dynamics: d = dt v 0 v 0 t Mehul Motani,

6 What is the Kalman Filter? Optimum Recursive Data Processing Algorithm Optimum: Uses all available data ML, Least Squares, MVUE Recursive: Does not store all data Critical for implementation Data Processing Incorporates discrete-time Mehul Motani, 000 Assumptions Linear (Discrete) System Model Admits tractable analysis Linear systems theory is quite thorough. White Noise Real systems are bandpass. Gaussian Noise Use Central Limit Theorem Mehul Motani, 000 6

7 General System Dynamic Model + = Φ = M + B + v w + u State Vector at time : R l Input Vector : w R Measurement Vector : R n m n n State Transition Matri : Φ n l Input Relation Matri : m n Measurement Matri : B M AWG Process Noise : u ~ Normal(0, Q) AWG Measurement Noise : v ~ Normal(0, Mehul Motani, B = R = 0 Kalman s Model + = Φ = M + u Weiner Problem Given the observed values{, = m},find an estimate ˆ which minimies the epected Mehul Motani,

8 Kalman s Solution Orthogonal Projection ˆ = E[,,, m Using the state representation, he derives a recursive optimum solution. We will not derive, but rather motivate the solution. Mehul Motani, A Subtle Distinction Estimate Estimate Error Estimate Error Covariance a priori ˆ e ˆ = P = T [ e ] E e a posteriori ˆ e = ˆ P = E e T [ e Mehul Motani,

9 Basic Operation of the Filter Time Update (Predict) Project current state and covariance forward to the net time step, i.e. compute the net a priori estimates. Measurement Update (Correct) Update the a priori quantities using noisy measurements, i.e. compute the a posteriori Mehul Motani, The Optimum Solution ˆ ( M ˆ ) = + K Choose K to minimie the a posteriori error covariance. A minimiing form for K is ˆ K T ( M P M + R ) T = P Mehul Motani,

10 A Closer Loo K T ( M P M + R ) T = P M Good Measurements As R 0, K M Bad Measurements As P 0, K Mehul Motani, Kalman Filter Algorithm Initialestimatesfor P and ˆ + + Predict. Project state ahead ˆ = Φ ˆ. Project error covariance P = Φ P Φ + Q T. Compute the Kalman Gain K = P M T ( M P M + R ) ( M ˆ ). Compute a posteriori estimate ˆ = ˆ + K Correct T 3. Update error covariance P = ( I K M ) Mehul Motani,

11 Fine Tuning the Kalman Filter Measurement Noise Covariance, Q Can tae offline samples and estimate Process Noise Covariance, R Not so clear how to estimate Both quantities can be time varying! Choosing Q and R is actually an Mehul Motani, 000 Eample: Lost in Space Spacecraft accelerating with random bursts from its thrusters. + T T / + = 0 = + v R = σ v Q = σ a T / 4 T / Mehul Motani, 000 a 3 T / T

12 Kalman Filter Performance σ 0ft., 0.5ft./sec v = σ a =, R = σ Mehul Motani, Kalman Filter Performance σ v = 0ft., σ a = 0.5ft./sec, R Mehul Motani, 000 4

13 Kalman Filter Performance σ v = 0ft., σ a = 0.5ft./sec, R = Mehul Motani, Applications Navigational and Guidance Systems Radar tracing and Sonar ranging Satellite orbit computations Active Noise Control Predictive tracing for virtual reality MMSE receiver is Kalman Mehul Motani,

14 Recall our Assumptions Linear Discrete System Model White Measurement and Process Noise Gaussian Measurement and Process Noise The Kalman Filter is the best Mehul Motani, Variations on a filter Discrete-Discrete Kalman Filter Π Continuous-Discrete Kalman Filter Etended Kalman Mehul Motani,

15 Continuous-Discrete Kalman System Model Continuous Model for dynamical system Discrete measurement equations Why? Fleibility Irregularly spaced measurements Use numerical integration (e.g. Runge-Kutta) to project states Mehul Motani, Non-linear Systems Can we rela the linearity assumption? Nonlinear stochastic difference equation + = f ( = h(, v, w ), u Mehul Motani,

16 The Etended Kalman Filter Linearie about the current mean and covariance using Taylor Series notions. = ~ A ( ˆ ) + W w = ~ + H ~ + U u ( ) Use Jacobians to project ahead and to relate measurement to Mehul Motani, Non-Gaussian Noise Kalman filter is no longer universally optimum. It is still the minimum variance estimator amongst all linear unbiased Mehul Motani,

17 What did they do before Kalman? Weiner filter Developed by Weiner at MIT in the 940s Analyes time series in the frequency domain Applies only to stationary problems There is much wor on etending Weiner s ideas to nonstationary Mehul Motani, Kalman vs. Weiner Kalman Filter applies to both stationary and nonstationary problems Implementation Issues Weiner filter operates on all data directly for each estimate Kalman filter recursively conditions current estimate on all past Mehul Motani,

18 All Roads Lead From Gauss since all our measurements and observations are nothing more than approimations to the truth, the same must be true of all calculations resting upon them, and the highest aim of all computations made concerning concrete phenomena must be to approimate, as nearly as practicable, to the truth. But this can be accomplished in no other way than by a suitable combination of more observations than the number absolutely requisite for the determination of the unnown quantities. This problem can only be properly undertaen when an approimate nowledge of the orbit has been already attained, which is afterwards to be corrected so as to satisfy all the observations in the most accurate manner possible. -- From Theory of the Motion of the Heavenly Bodies Moving about the Sun in Conic Sections, Gauss, Mehul Motani, Discussion Non-white measurement and process noise. Non-independent noise What if the statistics of the noise are unnown or vary rapidly? Efficient as it is, the Kalman filter is still not practical for high dimensional systems. Can you approimate the Kalman filter for large Mehul Motani,

19 One Last Thing to Thin About Suppose we want to construct a time history of the states given all the measurements, rather than estimating the state as measurements come in. Can we do better than the Kalman Mehul Motani,