Applications in Industry (Extended) Kalman Filter. Week Date Lecture Title
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1 hp://elec34.com Applicaions in Indusry (Eended) Kalman Filer 26 School of Informaion echnology and Elecrical Engineering a he Universiy of Queensland Lecure Schedule: Week Dae Lecure ile 29-Feb Inroducion 3-Mar Sysems Overview Mar Sysems as Maps & Signals as Vecors -Mar Daa Acquisiion & Sampling 4-Mar Sampling heory 7-Mar Anialiasing Filers 2-Mar Discree Sysem Analysis 24-Mar onvoluion Review 28-Mar Holiday 3-Mar Apr Frequency Response & Filer Analysis 7-Apr Filers -Apr Digial Filers 4-Apr Digial Filers 8-Apr Digial Windows 2-Apr FF 25-Apr Holiday 28-Apr Inroducion o Feedback onrol 3-May Holiday 5-May Feedback onrol & Regulaion 9-May Servoregulaion/PID 2-May Inroducion o (Digial) onrol May Digial onrol Design & Sae-Space 9-May Observabiliy onrollabiliy & Sabiliy of Digial Sysems Digial onrol Sysems: Shaping he 23-May Dynamic Response & Esimaion 26-May Applicaions in Indusry 3-May Sysem Idenificaion & Informaion heory 2-Jun Summary and ourse Review ELE 34: Sysems 26 May 26-2
2 Announcemens PS 3 Peer Review ompeiion he PS 3 Review wih he highes Liker Score Deadline for reviews: June 3 (:59 pm) Good reviews discussed June 2 nd Las Lecure Reward: 34 ELE 34 Review Lab ( Lab 5 ): Redo any aspecs of any of he labs Review course! Review 25 Final eam (which we will do on June 2 nd also) ELE 34: Sysems 26 May 26-3 Follow Along Reading: B. P. Lahi Signal processing and linear sysems 998 K52.9.L G. Franklin J. Powell M. Workman Digial onrol of Dynamic Sysems 99 J26.F72 99 [Available as UQ Ebook] oday Sae-space FPW haper 6 - Design of Digial onrol Sysems Using Sae-Space Mehods Lahi h. 2 (?) ime onsan and Rae of Informaion ransmission Informaion heory! ELE 34: Sysems 26 May
3 </assessable> WARNING: No assessable Nohing beyond his poin is on he eam. (ecep for he eam review ) Do no pay aenion. Do no aemp o learn. ELE 34: Sysems 26 May 26-5 Eample : ELE 34: Sysems 26 May
4 F 2 SS onrol anonical Form) ELE 34: Sysems 26 May 26-7 onrol anonical Form as a Block Diagram ELE 34: Sysems 26 May
5 Modal Form F is no he only way o f2ss Parial-fracion epansion of he sysem Sysem poles appear as diagonals of Am wo issues: he elemens of mari maybe comple if he poles are comple I is non-diagonal wih repeaed poles ELE 34: Sysems 26 May 26-9 Modal Form ELE 34: Sysems 26 May 26-5
6 Modal Form Block Diagram ELE 34: Sysems 26 May 26 - Malab s f2ss Y s Given: = 25.4s+5.8 U s s s s+5.8 Ge a sae space represenaion of his sysem Malab: Answer: ELE 34: Sysems 26 May
7 Eample : Invered Pendulum ELE 34: Sysems 26 May 26-3 Digial onrol Wikipedia ar and pole ELE 34: Sysems 26 May
8 Invered Pendulum ELE 34: Sysems 26 May 26-5 Invered Pendulum Equaions of Moion he equaions of moion of an invered pendulum (under a small angle approimaion) may be linearized as: θ = ω ω = θ = Q 2 θ + Pu Where: Q 2 M + m = g Ml P = Ml. If we furher assume uniy Ml (Ml ) hen P ELE 34: Sysems 26 May
9 Invered Pendulum Sae Space We hen selec a sae-vecor as: = θ hence = θ = ω ω ω ω Hence giving a sae-space model as: A = Q 2 B = he resolven of which is: Φ s = si A s s = Q 2 = s s 2 Q 2 Q 2 s And a sae-ransiion mari as: Φ = cosh Q sinh Q Q Q sinh Q cosh Q ELE 34: Sysems 26 May 26-7 ar & Pole in Sae-Space Wih Obsacles? Swing-up is a lile more han sabilizaion See also: MER422 uorial : hp://roboics.iee.uq.edu.au/~mer422/pl/-week-pendulum.pdf ELE 34: Sysems 26 May
10 ar & Pole in Sae-Space Swing-up is a lile more han sabilizaion ELE 34: Sysems 26 May 26-9 Eample 2: ommand Shaping ELE 34: Sysems 26 May 26-2
11 Eperimens: Scanning Over Obsacle ELE 34: Sysems 26 May 26-2 ommand Shaping ELE 34: Sysems 26 May 26-22
12 Velociy Velociy Robus onrol: ommand Shaping for Vibraion Reducion Inegraed Planner onroller ommand Shapping + Σ Error Regulaor Plan unning Sensor ELE 34: Sysems 26 May ommand Shaping Original velociy profile * ime ommand-shaped velociy profile Inpu shaper ime ime ELE 34: Sysems 26 May
13 3 ommand Shaping Zero Vibraion (ZV) Zero Vibraion and Derivaive (ZVD) 2 d i i K K K A 2 e K d d i i K K K K K A 2 ) ( ) ( 2 ) ( May 26 - ELE 34: Sysems 25 Eperimens: ommand Shaping 26 May 26 - ELE 34: Sysems 26
14 Esimaion: Ye anoher way o bea he noise ELE 34: Sysems 26 May Along muliple dimensions ELE 34: Sysems 26 May
15 Sae Space We collec our se of uncerain variables ino a vecor = [ 2 N ] he se of values ha migh ake on is ermed he sae space here is a single rue value for bu i is unknown ELE 34: Sysems 26 May Sae Space Dynamics ELE 34: Sysems 26 May
16 Measured versus rue Measuremen errors are ineviable So add Noise o Sae... Sae Dynamics becomes: an represen his as a Normal Disribuion ELE 34: Sysems 26 May 26-3 Recovering he ruh Numerous mehods ermed Esimaion because we are rying o esimae he ruh from he signal A sraegy discovered by Gauss Leas Squares in Mari Represenaion ELE 34: Sysems 26 May
17 Recovering he ruh: erminology ELE 34: Sysems 26 May General Problem ELE 34: Sysems 26 May
18 Duals and Dual erminology ELE 34: Sysems 26 May Esimaion Process in Picures ELE 34: Sysems 26 May
19 Kalman Filer Process ELE 34: Sysems 26 May KF Process in Equaions ELE 34: Sysems 26 May
20 KF onsideraions ELE 34: Sysems 26 May E: Kinemaic KF: racking onsider a Sysem wih onsan Acceleraion ELE 34: Sysems 26 May
21 In Summary KF: he rue sae () is separae from he measured (z) Les you combine prior conrols knowledge wih measuremens o filer signals and find he ruh I regulaes he covariance (P) As P is he scaer beween z and So if P hen z (measuremens ruh) EKF: akes a aylor series approimaion o ge a local F (and G and H ) ELE 34: Sysems 26 May 26-4 Esimaion: Bayesian Perspecive ELE 34: Sysems 26 May
22 Kalman Filering (Opimal) esimaion of he (hidden) sae of a linear dynamic process of which we obain noisy (parial) measuremens Eample: radar racking of an airplane. Wha is he sae of an airplane given noisy radar measuremens of he airplane s posiion? ELE 34: Sysems 26 May Model Discree ime seps coninuous sae-space (Hidden) sae: measuremen: y Airplane eample: Posiion speed and acceleraion y ~ ELE 34: Sysems 26 May
23 23 Dynamics and Observaion model Linear dynamics model describes relaion beween he sae and he ne sae and he observaion: Airplane eample (if process has ime-sep ): ) ( ~ ) ( ~ R N V Q N W A v w v w y 2 2 A 26 May 26 - ELE 34: Sysems 45 Le be a normal disribuion of he iniial sae hen every is a normal disribuion of hidden sae. Recursive definiion: And every Y is a normal disribuion of observaion y. Definiion: Goal of filering: compue condiional disribuion Normal disribuions ELE 34: Sysems 26 May W A V Y Y Y y y
24 Normal disribuion Because s and Y s are normal disribuions Y y Y y is also a normal disribuion Normal disribuion is fully specified by mean and covariance We denoe: s N E Y y Ys y s Y y Y y Var Y y Y y Problem reduces o compuing and P s N s ˆ s P s s s ELE 34: Sysems 26 May Recursive updae of sae Kalman filering algorihm: repea ime updae: from compue a priori disribuion + Measuremen updae: from + (and given y + ) compue a poseriori disribuion Y Y 2 Y 3 Y 4 Y 5 ELE 34: Sysems 26 May
25 25 ime updae From compue a priori disribuion + : So: Q A AP A N W A A W A N W A W A N W A ˆ Var Var E E Var E Q A AP P A ˆ ˆ 26 May 26 - ELE 34: Sysems 49 From + (and given y + ) compue ompue a priori disribuion of he observaion Y + from + : Measuremen updae ELE 34: Sysems 26 May 26-5 R P N V V N V V N V Y ˆ Var Var E E Var E
26 Measuremen updae (con d) 2. Look a join disribuion of + and Y + : Y E Var N E Y ov ˆ P N ˆ P ov Y Y Var Y P P R where ov Y ov V ov ovv Var P ELE 34: Sysems 26 May 26-5 Measuremen updae (con d) Recall ha if hen 2 Z Z2 N Z Z z N z 3. ompue + + = ( + Y + = y + ): N ˆ P P P Y y P R ˆ y P R P ELE 34: Sysems 26 May
27 Measuremen updae (con d): his can also (ofen) be wrien in erms of he Kalman gain mari: ˆ K P P ˆ P K P y K ˆ P R ELE 34: Sysems 26 May Iniializaion hoose disribuion of iniial sae by picking and P Sar wih measuremen updae given measuremen y hoice for Q and R (ideniy) small Q: dynamics rused more small R: measuremens rused more ELE 34: Sysems 26 May
28 28 I. Model: II. Algorihm: Repea ime updae: Measuremen updae: (Bayesian) Kalman Filer Summary ELE 34: Sysems 26 May Q A AP P A ˆ ˆ P K P P K R P P K ˆ ˆ ˆ y ) ( ~ ) ( ~ R N V Q N W A v w v w y ake Aways: Kalman filer can be used in real ime Use s as opimal esimae of sae a ime and use P as a measure of uncerainy. Eensions: Dynamic process wih known conrol inpu Non-linear dynamic process Kalman smoohing: compue opimal esimae of sae given all daa y y wih > (no real-ime). Auomaic parameer (Q and R) fiing using EM-algorihm (Bayesian) Kalman Filer Summary [II] ELE 34: Sysems 26 May 26-56
29 Ne ime Informaion heory & More! Review: haper 6 of FPW haper 3 of Lahi Deeper Pondering?? ELE 34: Sysems 26 May Final Eam Review June 26 From: 2-4 In: (???) Some Review Noes (from ourse ebooks) hp://roboics.iee.uq.edu.au /~elec34/ues.hml ELE 34: Sysems 26 May
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