Adaptive State Estimation Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018
|
|
- Penelope Reed
- 5 years ago
- Views:
Transcription
1 Adaptive State Estimation Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 218! Nonlinearity of adaptation! Parameter-adaptive filtering! Test for whiteness of the residual! Bias estimation! Noise-adaptive filtering! Multiple model estimation Copyright 218 by Robert Stengel. All rights reserved. For educational use only Nonlinearity of Adaptation Adaptation required if System parameters are unknown System structure is unknown Disturbance/measurement statistics are uncertain Adaptive estimators are fundamentally nonlinear, even if the system is linear Parameters to be estimated multiply the state Statistics are derived from measurement residuals Estimator gain depends on parameter estimates 2
2 Parameter Variation Approximated as Process Noise White noise disturbance input ( process noise ) is similar to random parameter variation w'!x = F( p)x + Gu + Lw " F( p o )x t + F ( p o ) p Δp t # F( p o )x + Gu + L w L Δp x p w t Δp t ; E w' x + Gu + Lw F p o Δp is an ( n n) matrix L Δp x t is n s = ; E w' w' T τ w t Δp t matrix { } = Q'δ ( t τ ) however, model is approximate and nonlinear 3 Parameter-Adaptive Estimation (Parameter Identification via Extended Kalman Filter) 4
3 Parameter-Dependent Linear System Linear system with parameterdependent sensitivity matrices x = F p x t + G( p)u + L p = Hx + n z t w t E w t n t = ; E w t n t w T ( τ ) n T ( τ ) = Q R δ t τ Parameter vector, p(t), could be Known: a prescribed function of time, and system is LTV Unknown: the output of a random dynamic process 5 Dynamic Model for Parameter Estimation Augment vector to include original state and parameter vector x A System is nonlinear: parameter is in the augmented state x A = z t F p t x t = H A p + G p f p x t p t x t p t u + L p w x f A x,p,u,w p,w p + n = H p x { } p + n + b p H p t : Unknown scale factors and coupling terms (TBD) b p t : Unknown bias error (TBD) BUT What is the model for time variation, p(t)? 6
4 Parameter Vector Must Have a Dynamic Model Unknown constant: p(t) = constant p = ; p( ) = p o ; P p ( ) = P po Random p(t) (integrated white noise) p = w p ; p( ) = ; P p ( ) = P po E w p = ; E w p w T p ( τ ) = Q pδ t τ Linear dynamic system (Markov process) p = Ap + Bw p f p p t,w p t ; w p t N,Q p 7 Dynamic Models for the Parameter Vector Doubly integrated white noise p M = p t p D = I p t p D + w p t p = p D Triply integrated white noise Parameter vector Parameter rate of change p M = p t p D p A = I I p t p D p A + p p D p A w p = Parameter vector Parameter rate of change Parameter acceleration 8
5 Integrated White Noise Models of a Parameter Third integral models slowly varying, smooth parameter Second integral is smoother but still has fast changes First integral of white noise has abrupt jumps, valleys, and peaks White noise 9 Hybrid Filter for Parameter Estimation P A t k ( ) ˆx A t k ( ) = x t A ( + ) k 1 + f A ˆx A ( τ ),u( τ ) dτ = P t A ( + ) k 1 + F A ( τ )P A τ = P A t k ( ) K t k Extrapolation of Augmented State t k t k 1 t k t k 1 Covariance Extrapolation H T A ( t k ) H A t k + P A τ F A T τ Filter Gain Calculation P A t k + L A τ H T A Q' C ( τ )L T A ( τ ) dτ ( t k ) + R t k 1 ( F A,H A ) must be locally observable except (perhaps) at isolated points 1
6 Hybrid Filter for Parameter Estimation ˆx A t k ( + ) = ˆx A t k P A t k + State Update { } + K ( t k ) z( t k ) h ˆx t k ( ) Covariance Update = I n K( t k )H A ( t k ) P A t k 11 Example: Estimation of Variable Rotary Load for a Robot Arm Elbow x 1 (t) x 2 (t) Parameter, p t Closed-loop dynamic equation = 1 c 1 / J t c 2 / J Rotary load inertia, J t x 1 (t) x 2 (t) + c 1 / J y c ; c 1,c 2 : Control gains (given) 12
7 x A = z = Augmented State and Measurement for Unknown Inertia Modeled as Doubly Integrated Parameter x 1 x 2 J J 1 1 x 1 x 2 p t p D x 1 x 2 p t Angle Angular Rate Rotary Load Inertia p D Inertia Rate + n 1 n 2 13 Nonlinear Dynamic Equation with Unknown Inertia Modeled as Doubly Integrated White Noise x A = x 1 x 2 p t p D + = c 1 / p t 1 1 c 1 / p t c 2 / p 1 y c + w 2 w pd t f A [ ] x 1 x 2 p t p D 14
8 Stability and Measurement Matrices F A = 1 1 c 1 2 ˆp 2 c ˆp c 1 / ˆp c 2 / ˆp H A = P A Means and Covariances ˆx A ( ) = E x 1 ( ) = E x A ( ) ˆx ( A ) ˆx 1 ( ) p( ) ˆp ( ) T { x A ( ) ˆx A ( ) } ŵ = ; Q = E w t ˆn = ; R = E n t w T ( τ ) n T ( τ ) 16
9 Hybrid Filter for Robot Elbow State and Load Estimation Extrapolation of Augmented State Covariance Extrapolation Convergence is problem-dependent Qualitative observability of parameter Actual and assumed uncertainty covariances Accuracy of dynamic model P A t k ( ) ˆx A t k ( ) = x t A ( + ) k 1 + f A ˆx A ( τ ),u( τ ) dτ = P A t k 1 ( + ) + F A ( τ )P A τ t k t k 1 + P A τ t k t k 1 F A T τ + L A τ Q' C ( τ )L T A ( τ ) dτ Filter Gain Calculation State Update Covariance Update = P A t k ( ) K t k ˆx A t k ( + ) H T A ( t k ) H A t k = ˆx A t k + K t k P A t k ( ) H T A ( t k ) + R t k z( t k ) h ˆx t k ( ) 1 { } P A t k ( + ) = I K t n ( k )H A ( t k ) P A t k ( ) 17 Weathervane Example (OCE, 4.7-1) True System!x 1!x 2 = z 1 z 2 = 1 1 x 1 x 2 Q = q = 1; R = x 1 x 2 + w 4 + n 1 n Assume F(2,1) is an unknown constant, a Linear Dynamic System for Covariance Estimation Δˆ! x1 Δˆ! x2 Δ ˆ! a = 1 â.8 ˆx 1 Δˆx 1 Δˆx 2 Δâ + â Δw 18
10 Weathervane Example (OCE, 4.7-1) State/Parameter Estimator ˆ!x 1 ˆ!x 2 ˆ!a ˆx 2 = âˆx 1.8 ˆx 2 ˆx 1 ˆx 2 â Initial Conditions for State/ Parameter Estimation = 4.4 k 11 k 12 + k 21 k 22 k 31 k 32 ; P = z 1 ˆx 1 z 1 ˆx 2 2! Parameter estimate approaches constant value as t, p 13, p 23, p 33 t p 33 â ˆx 1! Estimate may be biased 19 Bias Estimation and Noise-Adaptive Filtering 2
11 Residuals and Optimal Filtering Linear-optimal filtering has the innovations property r k a) Optimal estimation extracts all the available information from the measurements b) Optimal measurement residual should be zeromean white noise c) State estimate should be orthogonal to the error Residual and its statistics ( )! z k Hˆx k ( ) = H x k ( ) ˆx k ( ) + n k E r k! ˆr k E r k ( )r T k ( )! S k T E ˆx k ( )n k = E Φˆx k 1 + = = HP k H T + R k State estimate is orthogonal to the measurement noise T { + Γu k 1 n k } = 21 Residual Should Be White Noise if State Estimate is Optimal C( k) = Test for whiteness using normalized autocovariance function Sampled (batch process) estimate of the autocovariance function matrix, C(k) N 1 N 1 r n ( )r T n+k ( ), k (integer) << N n=k k dim C k = r r Normalize diagonal elements of C(k) by their zero-lag (k = ) values ρ ij ( k) = c ij k c ii 22
12 Test for Residual Whiteness If r k is white ρ ij ( k) = 1, i = j and k =, i j or k, N Off-diagonal terms should be negligible For finite sample, 95% confidence limit based on diagonal elements ρ ii ( k) 1.96 N, k Test is passed if 19 out of 2 non-zero-lag samples are within the limit Mehra, IEEE TAC, Bias Example: Advanced Dependent Surveillance (ADS-B) System for Air Traffic Control! Surveillance and tracking radars on ground! GPS/Inertial/Air data measurements in aircraft! Satellite/Line-of-sight communications links! Air traffic control centers (ATCC)! Prevent collisions! Maintain efficient flow of air traffic 24
13 Dynamics of Individual Aircraft Simplified surveillance equations of motion* Neglect fast (attitude) dynamics of aircraft x y z v x v y v z = x y z v x v y v z + w x w y w z * Latitude, longitude, and altitude rather than (x, y, z) 25 Air Traffic Control Center Measurements and Communicated Locations of Individual Aircraft Ground-based radar measurements GPS measurements z = Aircraft Inertial navigation measurements Air data measurements = ( Hx + n) + b b = Biases = Radar Range Bias Radar Azimuth Bias Aircraft Altimeter Bias ADS-B Longitude Bias ADS-B Latitude Bias ADS-B Altitude Bias ADS-B North Velocity Bias ADS-B East Velocity Bias Bias is a quasi-constant error that is not related to random noise Least-squares estimator does not reduce bias error 26
14 Measurement Bias and Covariance Estimation Batch processing estimates r ( ) ˆb ( ) = 1 N N i=1 r i ( ) = 1 N N i=1 z i Hˆx i ( ) Bias Estimate ˆR = 1 N ( r i r i )( r i r i ) T N 1 N 1 N HP i ( )H T i=1 = Ŝ HP av ( )H T, for large N Measurement Noise Estimate where Ŝ = 1 N 1 N i=1 ( r i r i ) r i r i T Sample Covariance Matrix 27 Running Estimate of Measurement Bias and Error Covariance { ˆb k 1 } ˆb k = ˆb k 1 + k bias z k Hˆx k ˆR k = ˆR k 1 + k noise = ˆb k 1 + k bias r k ˆb k 1 = [ 1 k bias ] ˆb k 1 + k bias r k { T HP ( k )H } T ( r k ˆb k ) r k ˆb k k bias, k noise < 1 ˆR k 1 Options! Choose ad hoc recursive (digital filter) gain! Use weighted least-squares estimator! Incorporate in an integrated parameter-adaptive filter 28
15 Disturbance Bias Estimation System equation x k +1 = Φx k + w k Disturbance residual w k = Φx k x k +1 Φˆx k + ˆx k +1 ( + ) w ŵ ( + ) = 1 N Sample mean N i=1 Φˆx i ( + ) ˆx i+1 ( + ) = 1 N N i=1 w i 29 Disturbance Covariance Estimation Ŵ = Sample covariance 1 N 1 N i=1 ( w i w) w i w T Disturbance covariance estimate ˆQ = Ŵ N 1 N ΦP av ( + )Φ T! Ŵ ΦP av ( + )Φ T 3
16 Running Estimate of Disturbance Bias and Covariance { } ŵ k ( + ) = ŵ k 1 ( + ) + k bias w k ( + ) ŵ k 1 ( + ) ˆQ k = ˆQ k 1 + k noise w k ŵ k ( w k ŵ k ) T N 1 k bias, k noise < 1 N ΦP k ( + )Φ T ˆQ k 1! Options as before! Choose add hoc recursive gain! Use weighted least-squares estimator! Incorporate in an integrated parameter-adaptive filter 31 Noise-and-Bias Adaptive Filter Use means and covariances from ad hoc estimation in Kalman filter ˆx k ( ) = Φ k 1 ˆx k 1 ( + ) + Γ k 1 u k 1 + ŵ k 1 ( + ) P k K k = P k T ( ) = Φ k 1 P k 1 ( + )Φ k 1 + ˆQ k 1 T ( )H k H k P k ˆx k ( + ) = ˆx k ( ) + K k z k H k ˆx k ( )H T k + ˆR k 1 + ˆb k = P 1 T k ( ) + H k ˆRk 1 H k P k + 1 or incorporate as parameters in a unified parameteradaptive filter 32
17 Multiple Model Estimation 33 Multiple Model Estimation! Bank of Kalman filters, each tuned to a different hypothesis! Different model parameters or structures! Different uncertainty models! Different initial conditions! Best performance determined by a hypothesis test, e.g., Maximum Likelihood! State estimate chosen accordingly 34
18 Hypothesis Testing 35 Multiple Model Estimation 36
19 Multiple Model Estimation! Consider J systems distinguished by J parameter vectors! Conditional probability mass function for the j th parameter set Pr( p j z k ) = I i=1 pr( z k p j )Pr( p j ) k 1 pr( z k p i )Pr p i k 1! Probability that the measurement at k 1 was obtained is one; therefore, Pr( p j ) k 1 = Pr( p j z k 1 )!... and the equation forms the basis for a recursion Pr( p j z k ) = J i=1 pr z k p j pr( z k p i )Pr p i k 1 Pr( p j z k 1 ) 37 Multiple Model Estimation Conditional probability density function z k must be found With the true parameter set z k = Hx k + n k x k = Φx k 1 + Γu k 1 + w k 1, x o = x pr z k p If the true state were known = pr z k x k ( p) 1 = ( 2π ) n/2 1/2 R e k 1 = ( 2π ) n/2 1/2 R e k 1 ( 2 z k Hx k ) T R 1 k ( z k Hx k ) 1 2 n k T R k 1 n k However, only an estimate of x k is available 38
20 Multiple Model Estimation! The state is estimated by a Kalman filter for the true parameter pr z k ˆx k ( p) = 1 2π n/2 S k 1/2 e 1 2 r k T S k 1 r k with r k ( ) = z k Hˆx k ( ) S k = HP k ( )H T + R k! The bank of J Kalman filters is formed with each filter assuming that different parameters are the true parameter 39 Conditional Probabilities and the Adaptive State Estimate Conditional probabilities for each hypothesis Pr( p j z k ) = J i=1 ( p j ) Pr ( p j z k 1 ) { pr z k ˆx k ( p i ) Pr( p i z k 1 )} pr z k ˆx k, j = 1, J A) State estimate: Estimate with highest conditional probability ˆx k ( + ) = ˆx k p k * [ ], p k * = argmax Pr( p i z k ), i = 1, J B) State estimate: Probability-weighted sum of the state estimates = Pr( p i z k ) ˆx k p i,( + ) ˆx k + J { } i=1 4
21 Weathervane Example (OCE, 4.7-2)! 2 nd -order system with three hypothesized natural frequencies ω 2 n = [ ] [ ] Correct 4.4 Expected ζ =.1! Option B: Algorithm searches, then homes in on correct solution 41 Caveats Each hypothesis requires a separate filter Number of required parallel filters may be large if all combinations are possible N # of filters = n i # of values, i = 1, N # of parameters i=1 True parameters may lie between tabulated values Solutions Pruning Sequentially eliminate least likely combinations Adaptive hypothesis adjustment ( parameter tracking ) Analogous to particle filtering; track most likely combinations, adjust distribution, drop low-probability hypotheses 42
22 Estimating Parameters of Nonlinear Systems Using Optimal Smoothing 43 Estimation Before Modeling* 1. Incorporate the unknown parameters in the state 2. Estimate the state using optimal smoother 1. Extended Kalman Filter (forward pass) 2. Modified Bryson-Frazier Smoother (backward pass) 3. Use multivariate regression to model aerodynamic coefficients as functions of state and control Measured vs. Estimated Yaw Rate 44
23 Smoothing vs. Filtering Filtering introduces a lag in the estimate The estimator is a Low-Pass Filter Causal, i.e., it uses measurements up to the current time but has no information about the future state Smoothing introduces no lag in the estimate The estimator is a Low-Pass Filter Non-Causal, i.e., it uses measurements before and after the current time Can Implement as forward and backward processing with lowpass filter H smoother H filter ( s) = ( s) = a s a, causal a 2 ( s a) s + a, non-causal * See Bryson & Frazier, 1963; Rauch, Tung, Streibel, 196;, Fraser, 1967; Bryson & Ho, 1969; Bierman, 1973, 1977; and papers on square-root implementations 45 Bryson-Frazier Optimal Smoother Smoothing viewed as deterministic optimal control problem * min J est = 1 x( t ), w, n 2 ΔxT (t )P 1 (t )Δx(t )+ 1 2 tf t w T (t ) n T (t ) Q 1 R 1 w(t ) n(t ) dt Subject to constraints Δx(t ) = x(t ) E [ x(t )]!x ( t ) = f x( t ), t + g x( t ), t w t = h x t z t, t + n(t ) Introduce adjoint vector, Hamiltonian, * Bryson, Frazier, 1963; Bryson, Ho,
24 Modified Bryson-Frazier Smoother * Motivated by Rauch-Tung-Streibel but viewed as recursive form of Bryson-Frazier Augment Hybrid Extended Kalman Filter results with backward pass through data from final to initial time Initialize adjoint variable and its covariance at final time, t f : λ f Λ f ( ) = H T f D 1 f ( )Δz f ( ) = H T f D 1 f ( )H f where T D f ( ) = H f P f ( )H f Δz f ( ) = z f h ˆx f ( ) * Bierman, Modified Bryson-Frazier Smoother Adjoint vector and covariance differential equations λ k Λ k!λ t = F T t = F T t!λ t λ ( t ) Λ t Λ T ( t ) F( t ) Propagate adjoint vector and covariance from final to initial condition λ k ( + ) = λ k+1 Λ k ( + ) = Λ k+1 Update adjoint vector and covariance ( ) = λ k + ( ) = I K k H k t k ( + ) F T t t k+1 ( ) t k ( + ) F T t t k+1 ( ) λ ( t )dt Λ t Λ T ( t ) F( t ) H k T D k 1 Δz k + D k K k T T Λ k+1 ( + ) I K k H k dt + H k T D k 1 H k 48
25 Modified Bryson-Frazier Smoother Smoothed estimates of state and covariance are corrections to filter estimates ˆx k + = ˆx k ( + ) P k ( + )λ k ( + ) ˆ P k ( + ) = P k ( + ) P k ( + )Λ k ( + )P k ( + ) 49 Schweizer 2-32 Aerodynamic Coefficients of a Sailplane from Flight Data* Princeton University Flight Research Laboratory * Sri-Jayantha, M., and Stengel, R., Determination of Aerodynamic Coefficients Using the Estimation-Before-Modeling Method, J. Aircraft, 25(9), 1988 [on Blackboard] 5
26 Dynamic Equations of the Sailplane (Attitude from 4-Element Quaternion Vector, q) Position!r I = H B I v B!q = Qq = fcn ω B Rotation Matrix and Inverse H B I, H I B are functions of q Velocity Quaternion!v B = 1 m F B + H I B g I "ω B v B Angular Rate!ω B = I B 1 M B "ω B I B ω B See MAE 331 Lecture 9 slides, 51 F B = I xx I xy I xz I B = I xy I yy I yz I xz I yz I zz X aero Y aero Z aero Inertia Matrix = B C X C Y C Z B Aircraft Characteristics Expressed in Body Frame of Reference 1 2 ρv 2 S B b = wing span c = mean aerodynamic chord S = wing area Aerodynamic force and moment vectors M B = L aero M aero N aero C l b = C m c C n b B B 1 2 ρv 2 S Force and moment coefficients modeled as triply integrated white noise 38-component state vector 52
27 Normal Force and Lift Coefficient Estimates Smoothed Estimate of Normal-Force Coefficient (C z ) History, including Dynamic Effects Binned Estimate of Lift Coefficient (C L ) Static Dependence on Angle of Attack Regression applied within angle-of-attack bins 53 Next Time: Stochastic Optimal Control 54
Nonlinear State Estimation! Particle, Sigma-Points Filters!
Nonlinear State Estimation! Particle, Sigma-Points Filters! Robert Stengel! Optimal Control and Estimation, MAE 546! Princeton University, 2017!! Particle filter!! Sigma-Points Unscented Kalman ) filter!!
More informationMiscellaneous. Regarding reading materials. Again, ask questions (if you have) and ask them earlier
Miscellaneous Regarding reading materials Reading materials will be provided as needed If no assigned reading, it means I think the material from class is sufficient Should be enough for you to do your
More information9 Multi-Model State Estimation
Technion Israel Institute of Technology, Department of Electrical Engineering Estimation and Identification in Dynamical Systems (048825) Lecture Notes, Fall 2009, Prof. N. Shimkin 9 Multi-Model State
More informationNonlinear State Estimation! Extended Kalman Filters!
Nonlinear State Estimation! Extended Kalman Filters! Robert Stengel! Optimal Control and Estimation, MAE 546! Princeton University, 2017!! Deformation of the probability distribution!! Neighboring-optimal
More informationStochastic Optimal Control!
Stochastic Control! Robert Stengel! Robotics and Intelligent Systems, MAE 345, Princeton University, 2015 Learning Objectives Overview of the Linear-Quadratic-Gaussian (LQG) Regulator Introduction to Stochastic
More informationTranslational and Rotational Dynamics!
Translational and Rotational Dynamics Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Copyright 217 by Robert Stengel. All rights reserved. For educational use only.
More informationStochastic and Adaptive Optimal Control
Stochastic and Adaptive Optimal Control Robert Stengel Optimal Control and Estimation, MAE 546 Princeton University, 2018! Nonlinear systems with random inputs and perfect measurements! Stochastic neighboring-optimal
More informationKalman Filter. Predict: Update: x k k 1 = F k x k 1 k 1 + B k u k P k k 1 = F k P k 1 k 1 F T k + Q
Kalman Filter Kalman Filter Predict: x k k 1 = F k x k 1 k 1 + B k u k P k k 1 = F k P k 1 k 1 F T k + Q Update: K = P k k 1 Hk T (H k P k k 1 Hk T + R) 1 x k k = x k k 1 + K(z k H k x k k 1 ) P k k =(I
More informationState-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Fin. Econometrics / 53
State-space Model Eduardo Rossi University of Pavia November 2014 Rossi State-space Model Fin. Econometrics - 2014 1 / 53 Outline 1 Motivation 2 Introduction 3 The Kalman filter 4 Forecast errors 5 State
More informationSequential State Estimation (Crassidas and Junkins, Chapter 5)
Sequential State Estimation (Crassidas and Junkins, Chapter 5) Please read: 5.1, 5.3-5.6 5.3 The Discrete-Time Kalman Filter The discrete-time Kalman filter is used when the dynamics and measurements are
More information1 Kalman Filter Introduction
1 Kalman Filter Introduction You should first read Chapter 1 of Stochastic models, estimation, and control: Volume 1 by Peter S. Maybec (available here). 1.1 Explanation of Equations (1-3) and (1-4) Equation
More informationEvaluation of different wind estimation methods in flight tests with a fixed-wing UAV
Evaluation of different wind estimation methods in flight tests with a fixed-wing UAV Julian Sören Lorenz February 5, 2018 Contents 1 Glossary 2 2 Introduction 3 3 Tested algorithms 3 3.1 Unfiltered Method
More informationStochastic Models, Estimation and Control Peter S. Maybeck Volumes 1, 2 & 3 Tables of Contents
Navtech Part #s Volume 1 #1277 Volume 2 #1278 Volume 3 #1279 3 Volume Set #1280 Stochastic Models, Estimation and Control Peter S. Maybeck Volumes 1, 2 & 3 Tables of Contents Volume 1 Preface Contents
More informationLecture 2: From Linear Regression to Kalman Filter and Beyond
Lecture 2: From Linear Regression to Kalman Filter and Beyond January 18, 2017 Contents 1 Batch and Recursive Estimation 2 Towards Bayesian Filtering 3 Kalman Filter and Bayesian Filtering and Smoothing
More informationA Blade Element Approach to Modeling Aerodynamic Flight of an Insect-scale Robot
A Blade Element Approach to Modeling Aerodynamic Flight of an Insect-scale Robot Taylor S. Clawson, Sawyer B. Fuller Robert J. Wood, Silvia Ferrari American Control Conference Seattle, WA May 25, 2016
More informationChapter 4 The Equations of Motion
Chapter 4 The Equations of Motion Flight Mechanics and Control AEM 4303 Bérénice Mettler University of Minnesota Feb. 20-27, 2013 (v. 2/26/13) Bérénice Mettler (University of Minnesota) Chapter 4 The Equations
More informationLecture AC-1. Aircraft Dynamics. Copy right 2003 by Jon at h an H ow
Lecture AC-1 Aircraft Dynamics Copy right 23 by Jon at h an H ow 1 Spring 23 16.61 AC 1 2 Aircraft Dynamics First note that it is possible to develop a very good approximation of a key motion of an aircraft
More informationOptimization-Based Control
Optimization-Based Control Richard M. Murray Control and Dynamical Systems California Institute of Technology DRAFT v1.7a, 19 February 2008 c California Institute of Technology All rights reserved. This
More informationOptimal Control and Estimation MAE 546, Princeton University Robert Stengel, Preliminaries!
Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, 2018 Copyright 2018 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/mae546.html
More informationTTK4190 Guidance and Control Exam Suggested Solution Spring 2011
TTK4190 Guidance and Control Exam Suggested Solution Spring 011 Problem 1 A) The weight and buoyancy of the vehicle can be found as follows: W = mg = 15 9.81 = 16.3 N (1) B = 106 4 ( ) 0.6 3 3 π 9.81 =
More informationECE276A: Sensing & Estimation in Robotics Lecture 10: Gaussian Mixture and Particle Filtering
ECE276A: Sensing & Estimation in Robotics Lecture 10: Gaussian Mixture and Particle Filtering Lecturer: Nikolay Atanasov: natanasov@ucsd.edu Teaching Assistants: Siwei Guo: s9guo@eng.ucsd.edu Anwesan Pal:
More informationPrinciples of the Global Positioning System Lecture 11
12.540 Principles of the Global Positioning System Lecture 11 Prof. Thomas Herring http://geoweb.mit.edu/~tah/12.540 Statistical approach to estimation Summary Look at estimation from statistical point
More informationOnline monitoring of MPC disturbance models using closed-loop data
Online monitoring of MPC disturbance models using closed-loop data Brian J. Odelson and James B. Rawlings Department of Chemical Engineering University of Wisconsin-Madison Online Optimization Based Identification
More informationFactor Analysis and Kalman Filtering (11/2/04)
CS281A/Stat241A: Statistical Learning Theory Factor Analysis and Kalman Filtering (11/2/04) Lecturer: Michael I. Jordan Scribes: Byung-Gon Chun and Sunghoon Kim 1 Factor Analysis Factor analysis is used
More informationLinear-Optimal State Estimation
Linear-Optimal State Estimation Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 218 Linear-optimal Gaussian estimator for discrete-time system (Kalman filter) 2 nd -order example
More informationSmoothers: Types and Benchmarks
Smoothers: Types and Benchmarks Patrick N. Raanes Oxford University, NERSC 8th International EnKF Workshop May 27, 2013 Chris Farmer, Irene Moroz Laurent Bertino NERSC Geir Evensen Abstract Talk builds
More informationMinimization of Static! Cost Functions!
Minimization of Static Cost Functions Robert Stengel Optimal Control and Estimation, MAE 546, Princeton University, 2017 J = Static cost function with constant control parameter vector, u Conditions for
More informationKalman filtering and friends: Inference in time series models. Herke van Hoof slides mostly by Michael Rubinstein
Kalman filtering and friends: Inference in time series models Herke van Hoof slides mostly by Michael Rubinstein Problem overview Goal Estimate most probable state at time k using measurement up to time
More informationOPTIMAL ESTIMATION of DYNAMIC SYSTEMS
CHAPMAN & HALL/CRC APPLIED MATHEMATICS -. AND NONLINEAR SCIENCE SERIES OPTIMAL ESTIMATION of DYNAMIC SYSTEMS John L Crassidis and John L. Junkins CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London
More informationTime Response of Dynamic Systems! Multi-Dimensional Trajectories Position, velocity, and acceleration are vectors
Time Response of Dynamic Systems Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 217 Multi-dimensional trajectories Numerical integration Linear and nonlinear systems Linearization
More informationThe Kalman Filter ImPr Talk
The Kalman Filter ImPr Talk Ged Ridgway Centre for Medical Image Computing November, 2006 Outline What is the Kalman Filter? State Space Models Kalman Filter Overview Bayesian Updating of Estimates Kalman
More informationState-space Model. Eduardo Rossi University of Pavia. November Rossi State-space Model Financial Econometrics / 49
State-space Model Eduardo Rossi University of Pavia November 2013 Rossi State-space Model Financial Econometrics - 2013 1 / 49 Outline 1 Introduction 2 The Kalman filter 3 Forecast errors 4 State smoothing
More informationAlternative Expressions for the Velocity Vector Velocity restricted to the vertical plane. Longitudinal Equations of Motion
Linearized Longitudinal Equations of Motion Robert Stengel, Aircraft Flig Dynamics MAE 33, 008 Separate solutions for nominal and perturbation flig paths Assume that nominal path is steady and in the vertical
More informationLecture 7: Optimal Smoothing
Department of Biomedical Engineering and Computational Science Aalto University March 17, 2011 Contents 1 What is Optimal Smoothing? 2 Bayesian Optimal Smoothing Equations 3 Rauch-Tung-Striebel Smoother
More informationOptimal Control and Estimation MAE 546, Princeton University Robert Stengel, Preliminaries!
Optimal Control and Estimation MAE 546, Princeton University Robert Stengel, 2017 Copyright 2017 by Robert Stengel. All rights reserved. For educational use only. http://www.princeton.edu/~stengel/mae546.html
More informationLinearized Equations of Motion!
Linearized Equations of Motion Robert Stengel, Aircraft Flight Dynamics MAE 331, 216 Learning Objectives Develop linear equations to describe small perturbational motions Apply to aircraft dynamic equations
More informationLecture 6: Bayesian Inference in SDE Models
Lecture 6: Bayesian Inference in SDE Models Bayesian Filtering and Smoothing Point of View Simo Särkkä Aalto University Simo Särkkä (Aalto) Lecture 6: Bayesian Inference in SDEs 1 / 45 Contents 1 SDEs
More informationMini-Course 07 Kalman Particle Filters. Henrique Massard da Fonseca Cesar Cunha Pacheco Wellington Bettencurte Julio Dutra
Mini-Course 07 Kalman Particle Filters Henrique Massard da Fonseca Cesar Cunha Pacheco Wellington Bettencurte Julio Dutra Agenda State Estimation Problems & Kalman Filter Henrique Massard Steady State
More informationLinear-Quadratic-Gaussian Controllers!
Linear-Quadratic-Gaussian Controllers! Robert Stengel! Optimal Control and Estimation MAE 546! Princeton University, 2017!! LTI dynamic system!! Certainty Equivalence and the Separation Theorem!! Asymptotic
More informationTime-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2015
Time-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 15 Asymptotic approach from time-varying to constant gains Elimination of cross weighting
More informationApplication of state observers in attitude estimation using low-cost sensors
Application of state observers in attitude estimation using low-cost sensors Martin Řezáč Czech Technical University in Prague, Czech Republic March 26, 212 Introduction motivation for inertial estimation
More informationModern Navigation. Thomas Herring
12.215 Modern Navigation Thomas Herring Estimation methods Review of last class Restrict to basically linear estimation problems (also non-linear problems that are nearly linear) Restrict to parametric,
More informationKalman Filters with Uncompensated Biases
Kalman Filters with Uncompensated Biases Renato Zanetti he Charles Stark Draper Laboratory, Houston, exas, 77058 Robert H. Bishop Marquette University, Milwaukee, WI 53201 I. INRODUCION An underlying assumption
More informationOrganization. I MCMC discussion. I project talks. I Lecture.
Organization I MCMC discussion I project talks. I Lecture. Content I Uncertainty Propagation Overview I Forward-Backward with an Ensemble I Model Reduction (Intro) Uncertainty Propagation in Causal Systems
More informationTime-Invariant Linear Quadratic Regulators!
Time-Invariant Linear Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 17 Asymptotic approach from time-varying to constant gains Elimination of cross weighting
More informationwith Application to Autonomous Vehicles
Nonlinear with Application to Autonomous Vehicles (Ph.D. Candidate) C. Silvestre (Supervisor) P. Oliveira (Co-supervisor) Institute for s and Robotics Instituto Superior Técnico Portugal January 2010 Presentation
More informationStatistics Homework #4
Statistics 910 1 Homework #4 Chapter 6, Shumway and Stoffer These are outlines of the solutions. If you would like to fill in other details, please come see me during office hours. 6.1 State-space representation
More informationRobot Control Basics CS 685
Robot Control Basics CS 685 Control basics Use some concepts from control theory to understand and learn how to control robots Control Theory general field studies control and understanding of behavior
More informationENGR352 Problem Set 02
engr352/engr352p02 September 13, 2018) ENGR352 Problem Set 02 Transfer function of an estimator 1. Using Eq. (1.1.4-27) from the text, find the correct value of r ss (the result given in the text is incorrect).
More informationFirst-Order Low-Pass Filter
Filters, Cost Functions, and Controller Structures Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 218! Dynamic systems as low-pass filters! Frequency response of dynamic systems!
More informationPROBABILISTIC REASONING OVER TIME
PROBABILISTIC REASONING OVER TIME In which we try to interpret the present, understand the past, and perhaps predict the future, even when very little is crystal clear. Outline Time and uncertainty Inference:
More information2D Image Processing. Bayes filter implementation: Kalman filter
2D Image Processing Bayes filter implementation: Kalman filter Prof. Didier Stricker Kaiserlautern University http://ags.cs.uni-kl.de/ DFKI Deutsches Forschungszentrum für Künstliche Intelligenz http://av.dfki.de
More informationAutomated Tuning of the Nonlinear Complementary Filter for an Attitude Heading Reference Observer
Automated Tuning of the Nonlinear Complementary Filter for an Attitude Heading Reference Observer Oscar De Silva, George K.I. Mann and Raymond G. Gosine Faculty of Engineering and Applied Sciences, Memorial
More informationSTATISTICAL ORBIT DETERMINATION
STATISTICAL ORBIT DETERMINATION Satellite Tracking Example of SNC and DMC ASEN 5070 LECTURE 6 4.08.011 1 We will develop a simple state noise compensation (SNC) algorithm. This algorithm adds process noise
More informationAdvanced Computational Methods in Statistics: Lecture 5 Sequential Monte Carlo/Particle Filtering
Advanced Computational Methods in Statistics: Lecture 5 Sequential Monte Carlo/Particle Filtering Axel Gandy Department of Mathematics Imperial College London http://www2.imperial.ac.uk/~agandy London
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY & Contents PREFACE xiii 1 1.1. 1.2. Difference Equations First-Order Difference Equations 1 /?th-order Difference
More informationProbabilistic Graphical Models
Probabilistic Graphical Models Brown University CSCI 2950-P, Spring 2013 Prof. Erik Sudderth Lecture 12: Gaussian Belief Propagation, State Space Models and Kalman Filters Guest Kalman Filter Lecture by
More informationArtificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik. Robot Dynamics. Dr.-Ing. John Nassour J.
Artificial Intelligence & Neuro Cognitive Systems Fakultät für Informatik Robot Dynamics Dr.-Ing. John Nassour 25.1.218 J.Nassour 1 Introduction Dynamics concerns the motion of bodies Includes Kinematics
More informationTime Response of Linear, Time-Invariant (LTI) Systems Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018
Time Response of Linear, Time-Invariant LTI Systems Robert Stengel, Aircraft Flight Dynamics MAE 331, 2018 Learning Objectives Methods of time-domain analysis Continuous- and discrete-time models Transient
More informationTime Series Analysis. James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY
Time Series Analysis James D. Hamilton PRINCETON UNIVERSITY PRESS PRINCETON, NEW JERSEY PREFACE xiii 1 Difference Equations 1.1. First-Order Difference Equations 1 1.2. pth-order Difference Equations 7
More information2D Image Processing. Bayes filter implementation: Kalman filter
2D Image Processing Bayes filter implementation: Kalman filter Prof. Didier Stricker Dr. Gabriele Bleser Kaiserlautern University http://ags.cs.uni-kl.de/ DFKI Deutsches Forschungszentrum für Künstliche
More informationCOMBINED ADAPTIVE CONTROLLER FOR UAV GUIDANCE
COMBINED ADAPTIVE CONTROLLER FOR UAV GUIDANCE B.R. Andrievsky, A.L. Fradkov Institute for Problems of Mechanical Engineering of Russian Academy of Sciences 61, Bolshoy av., V.O., 199178 Saint Petersburg,
More informationClassical Numerical Methods to Solve Optimal Control Problems
Lecture 26 Classical Numerical Methods to Solve Optimal Control Problems Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Necessary Conditions
More informationControl Systems! Copyright 2017 by Robert Stengel. All rights reserved. For educational use only.
Control Systems Robert Stengel Robotics and Intelligent Systems MAE 345, Princeton University, 2017 Analog vs. digital systems Continuous- and Discretetime Dynamic Models Frequency Response Transfer Functions
More information4 Derivations of the Discrete-Time Kalman Filter
Technion Israel Institute of Technology, Department of Electrical Engineering Estimation and Identification in Dynamical Systems (048825) Lecture Notes, Fall 2009, Prof N Shimkin 4 Derivations of the Discrete-Time
More informationDepartment of Aerospace Engineering and Mechanics University of Minnesota Written Preliminary Examination: Control Systems Friday, April 9, 2010
Department of Aerospace Engineering and Mechanics University of Minnesota Written Preliminary Examination: Control Systems Friday, April 9, 2010 Problem 1: Control of Short Period Dynamics Consider the
More informationNew Introduction to Multiple Time Series Analysis
Helmut Lütkepohl New Introduction to Multiple Time Series Analysis With 49 Figures and 36 Tables Springer Contents 1 Introduction 1 1.1 Objectives of Analyzing Multiple Time Series 1 1.2 Some Basics 2
More informationLecture 2: From Linear Regression to Kalman Filter and Beyond
Lecture 2: From Linear Regression to Kalman Filter and Beyond Department of Biomedical Engineering and Computational Science Aalto University January 26, 2012 Contents 1 Batch and Recursive Estimation
More informationStatistics 910, #15 1. Kalman Filter
Statistics 910, #15 1 Overview 1. Summary of Kalman filter 2. Derivations 3. ARMA likelihoods 4. Recursions for the variance Kalman Filter Summary of Kalman filter Simplifications To make the derivations
More informationKalman-Filter-Based Time-Varying Parameter Estimation via Retrospective Optimization of the Process Noise Covariance
2016 American Control Conference (ACC) Boston Marriott Copley Place July 6-8, 2016. Boston, MA, USA Kalman-Filter-Based Time-Varying Parameter Estimation via Retrospective Optimization of the Process Noise
More informationProblem 1: Ship Path-Following Control System (35%)
Problem 1: Ship Path-Following Control System (35%) Consider the kinematic equations: Figure 1: NTNU s research vessel, R/V Gunnerus, and Nomoto model: T ṙ + r = Kδ (1) with T = 22.0 s and K = 0.1 s 1.
More informationAircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2018
Aircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2018 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2018 by Robert Stengel. All rights reserved. For
More informationKalman Filter. Man-Wai MAK
Kalman Filter Man-Wai MAK Dept. of Electronic and Information Engineering, The Hong Kong Polytechnic University enmwmak@polyu.edu.hk http://www.eie.polyu.edu.hk/ mwmak References: S. Gannot and A. Yeredor,
More informationECE531 Lecture 11: Dynamic Parameter Estimation: Kalman-Bucy Filter
ECE531 Lecture 11: Dynamic Parameter Estimation: Kalman-Bucy Filter D. Richard Brown III Worcester Polytechnic Institute 09-Apr-2009 Worcester Polytechnic Institute D. Richard Brown III 09-Apr-2009 1 /
More informationPrediction of ESTSP Competition Time Series by Unscented Kalman Filter and RTS Smoother
Prediction of ESTSP Competition Time Series by Unscented Kalman Filter and RTS Smoother Simo Särkkä, Aki Vehtari and Jouko Lampinen Helsinki University of Technology Department of Electrical and Communications
More informationStochastic Processes. M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno
Stochastic Processes M. Sami Fadali Professor of Electrical Engineering University of Nevada, Reno 1 Outline Stochastic (random) processes. Autocorrelation. Crosscorrelation. Spectral density function.
More informationLinear Dynamical Systems
Linear Dynamical Systems Sargur N. srihari@cedar.buffalo.edu Machine Learning Course: http://www.cedar.buffalo.edu/~srihari/cse574/index.html Two Models Described by Same Graph Latent variables Observations
More informationProbabilistic Fundamentals in Robotics. DAUIN Politecnico di Torino July 2010
Probabilistic Fundamentals in Robotics Gaussian Filters Basilio Bona DAUIN Politecnico di Torino July 2010 Course Outline Basic mathematical framework Probabilistic models of mobile robots Mobile robot
More informationCALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems. CDS 110b
CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 110b R. M. Murray Kalman Filters 14 January 2007 Reading: This set of lectures provides a brief introduction to Kalman filtering, following
More informationECE 636: Systems identification
ECE 636: Systems identification Lectures 3 4 Random variables/signals (continued) Random/stochastic vectors Random signals and linear systems Random signals in the frequency domain υ ε x S z + y Experimental
More informationAircraft Flight Dynamics!
Aircraft Flight Dynamics Robert Stengel MAE 331, Princeton University, 2016 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2016 by Robert Stengel. All rights reserved. For
More informationSpace Surveillance with Star Trackers. Part II: Orbit Estimation
AAS -3 Space Surveillance with Star Trackers. Part II: Orbit Estimation Ossama Abdelkhalik, Daniele Mortari, and John L. Junkins Texas A&M University, College Station, Texas 7783-3 Abstract The problem
More informationA new unscented Kalman filter with higher order moment-matching
A new unscented Kalman filter with higher order moment-matching KSENIA PONOMAREVA, PARESH DATE AND ZIDONG WANG Department of Mathematical Sciences, Brunel University, Uxbridge, UB8 3PH, UK. Abstract This
More informationDynamics. 1 Copyright c 2015 Roderic Grupen
Dynamics The branch of physics that treats the action of force on bodies in motion or at rest; kinetics, kinematics, and statics, collectively. Websters dictionary Outline Conservation of Momentum Inertia
More informationState Estimation using Moving Horizon Estimation and Particle Filtering
State Estimation using Moving Horizon Estimation and Particle Filtering James B. Rawlings Department of Chemical and Biological Engineering UW Math Probability Seminar Spring 2009 Rawlings MHE & PF 1 /
More informationReturn Difference Function and Closed-Loop Roots Single-Input/Single-Output Control Systems
Spectral Properties of Linear- Quadratic Regulators Robert Stengel Optimal Control and Estimation MAE 546 Princeton University, 2018! Stability margins of single-input/singleoutput (SISO) systems! Characterizations
More informationDynamic data processing recursive least-squares
Dynamic data processing recursive least-squares Dynamic data processing recursive least-squares Delft University of Technology Department of Mathematical Geodesy and Positioning VSSD Series on Mathematical
More informationROBOTICS 01PEEQW. Basilio Bona DAUIN Politecnico di Torino
ROBOTICS 01PEEQW Basilio Bona DAUIN Politecnico di Torino Probabilistic Fundamentals in Robotics Gaussian Filters Course Outline Basic mathematical framework Probabilistic models of mobile robots Mobile
More informationCALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems. CDS 110b
CALIFORNIA INSTITUTE OF TECHNOLOGY Control and Dynamical Systems CDS 110b R. M. Murray Kalman Filters 25 January 2006 Reading: This set of lectures provides a brief introduction to Kalman filtering, following
More informationActive Flutter Control using an Adjoint Method
44th AIAA Aerospace Sciences Meeting and Exhibit 9-12 January 26, Reno, Nevada AIAA 26-844 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 9 12 Jan, 26. Active Flutter Control using an
More informationNonlinear Estimation Techniques for Impact Point Prediction of Ballistic Targets
Nonlinear Estimation Techniques for Impact Point Prediction of Ballistic Targets J. Clayton Kerce a, George C. Brown a, and David F. Hardiman b a Georgia Tech Research Institute, Georgia Institute of Technology,
More informationBasic Concepts in Data Reconciliation. Chapter 6: Steady-State Data Reconciliation with Model Uncertainties
Chapter 6: Steady-State Data with Model Uncertainties CHAPTER 6 Steady-State Data with Model Uncertainties 6.1 Models with Uncertainties In the previous chapters, the models employed in the DR were considered
More informationEM-algorithm for Training of State-space Models with Application to Time Series Prediction
EM-algorithm for Training of State-space Models with Application to Time Series Prediction Elia Liitiäinen, Nima Reyhani and Amaury Lendasse Helsinki University of Technology - Neural Networks Research
More informationRobotics & Automation. Lecture 25. Dynamics of Constrained Systems, Dynamic Control. John T. Wen. April 26, 2007
Robotics & Automation Lecture 25 Dynamics of Constrained Systems, Dynamic Control John T. Wen April 26, 2007 Last Time Order N Forward Dynamics (3-sweep algorithm) Factorization perspective: causal-anticausal
More informationMachine Learning 4771
Machine Learning 4771 Instructor: ony Jebara Kalman Filtering Linear Dynamical Systems and Kalman Filtering Structure from Motion Linear Dynamical Systems Audio: x=pitch y=acoustic waveform Vision: x=object
More informationEnvelopes for Flight Through Stochastic Gusts
AIAA Atmospheric Flight Mechanics Conference 08-11 August 2011, Portland, Oregon AIAA 2011-6213 Envelopes for Flight Through Stochastic Gusts Johnhenri R. Richardson, Ella M. Atkins, Pierre T. Kabamba,
More informationKalman filter using the orthogonality principle
Appendix G Kalman filter using the orthogonality principle This appendix presents derivation steps for obtaining the discrete Kalman filter equations using a method based on the orthogonality principle.
More informationECONOMETRIC METHODS II: TIME SERIES LECTURE NOTES ON THE KALMAN FILTER. The Kalman Filter. We will be concerned with state space systems of the form
ECONOMETRIC METHODS II: TIME SERIES LECTURE NOTES ON THE KALMAN FILTER KRISTOFFER P. NIMARK The Kalman Filter We will be concerned with state space systems of the form X t = A t X t 1 + C t u t 0.1 Z t
More informationLinear Discrete-time State Space Realization of a Modified Quadruple Tank System with State Estimation using Kalman Filter
Journal of Physics: Conference Series PAPER OPEN ACCESS Linear Discrete-time State Space Realization of a Modified Quadruple Tank System with State Estimation using Kalman Filter To cite this article:
More informationFrom Bayes to Extended Kalman Filter
From Bayes to Extended Kalman Filter Michal Reinštein Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception http://cmp.felk.cvut.cz/
More information