# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

13 Pages

### hamilton_and_lag

Course: MECHANICAL 2.141, Fall 2006
School: MIT
Rating:

Word Count: 1083

#### Document Preview

MECHANICAL NONLINEAR SYSTEMS LAGRANGIAN AND HAMILTONIAN FORMULATIONS Lagrangian formulation 1 Ek*(f,q) = 2 ft I(q) f q f generalized coordinates (displacement) generalized velocity (flow) Ek*(f,q) kinetic co-energy I(q) a configuration-dependent inertia tensor (matrix) Note: kinetic co-energy is a quadratic form in flow (generalized velocity). (Euler-)Lagrange equation: d Ek * Ek * dt f q = e Mod....

Register Now

#### Unformatted Document Excerpt

Coursehero >> Massachusetts >> MIT >> MECHANICAL 2.141

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
MECHANICAL NONLINEAR SYSTEMS LAGRANGIAN AND HAMILTONIAN FORMULATIONS Lagrangian formulation 1 Ek*(f,q) = 2 ft I(q) f q f generalized coordinates (displacement) generalized velocity (flow) Ek*(f,q) kinetic co-energy I(q) a configuration-dependent inertia tensor (matrix) Note: kinetic co-energy is a quadratic form in flow (generalized velocity). (Euler-)Lagrange equation: d Ek * Ek * dt f q = e Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 1 General form: df I(q) dt C(f,q) = e e generalized force (effort) C(f,q) contains coriolis and centrifugal forces Explicit state-determined form: dq dt = f df -1 dt = I(q) (e + C(f,q)) Note: The inertia tensor (matrix) must be inverted to find a statedetermined form. Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 2 Lagranges equation may include conservative generalized forces. e = econservative + enon-conservative Ep(q) econservative = q Ep(q) potential energy function d Ek* Ek* Ep(q) dt f q + q = enon-conservative Potential energy is not a function of generalized velocity. d L L f q = enon-conservative dt L(f,q) is the Lagrangian state function L(f,q) = Ek*(f,q) Ep(q) In the usual notation, Ek*(f,q) is written as T(f,q) and Ep(q) is written as V(q), hence L(f,q) = T(f,q) V(q) i.e., L=TV Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 3 Hamiltonian formulation Interaction between a capacitor and an inertia is the archetypal Hamiltonian system: dp/dt = H(p,q)/q q p dq/dt = H(p,q)/p generalized coordinates generalized momenta H(p,q) is the Hamiltonian state function. H(p,q) = Ek(p,q) + Ep(q) In this case H(p,q) is equal to the total energy in the system. Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 4 Lagrangian and Hamiltonian forms are related by a Legendre transformation. Lagrange used kinetic co-energy Hamilton used kinetic energy H(p,q) = ptf L(f,q) L(f,q) = ptf H(p,q) Differentiate with respect to generalized momentum. L(f,q)/p = 0 = f H(p,q)/p thus dq/dt = H(p,q)/p Generalized momentum is the gradient of kinetic co-energy with respect to flow (velocity) p = Ek*(f,q)/f = L(f,q)/f Lagrange's equation: dp/dt L(f,q)/q = e L(f,q)/q = H(p,q)/q thus dp/dt = H(p,q)/q + e These are Hamiltons equations dq/dt = H(p,q)/p dp/dt = H(p,q)/q + e Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 5 EXAMPLE: MULTIPLE DEGREE OF FREEDOM MECHANISM Generalized coordinates: q Generalized momentum p = Ek*(f,q)/f = I(q) f Kinetic co-energy is a positive definite quadratic form,. The inertia tensor is symmetric and real. Kinetic energy 1 Ek(p,q) = ptf Ek*(f,q) = ptf 2 ftI(q)f 1 Ek(p,q) = ptI(q)-1p 2 ptI(q)-1I(q)I(q)-1p 1 Ek(p,q) = 2 ptI(q)-1p Note: Kinetic energy is always a quadratic form in generalized momentum. Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 6 Hamiltons equations 1 dp/dt = [2 ptI(q)-1p]/q + e dq/dt = I(q)-1p Note: The Hamiltonian form may include arbitrary generalized forces (or torques) including dissipative or source terms. The most general form: dp/dt = H(p,q)/q + e(p,q,t) dq/dt = H(p,q)/p Sim. f(p,q,t) Mod. Dyn. Sys. Lagrangian and Hamiltonian forms page 7 Hamiltons formulation applies to other energy domains EXAMPLE: SIMPLE ELECTRIC CIRCUIT. The energy storage elements may be distinguished from the rest of the system: Generalized coordinates: charge, q, and flux linkage, Hamiltonian: H(q,) = q2/2C + 2/2L Again, the Hamiltonian is the total system energy. Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 8 Gradients: H/q = q/C H/ = /L Without the source or resistors, the conservative Hamiltonian equations would be eL = d/dt = H/q = q/C iC = dq/dt = H/ = /L With the source and resistors, the non-conservative Hamiltonian equations are d/dt = H/q + E0 eR2 dq/dt = H/ iR1 Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 9 CANONICAL TRANSFORMATIONS Choice of (state) variables is important in physical system modeling particularly important for nonlinear mechanical systems Geometry is fundamental. The choice of coordinates used to represent the geometry and kinematics of a system has a profound effect on the structure and complexity of its describing equations. Transformations of state variables are used extensively to analyze linear state determined systems. e.g., physical variables to diagonal form Structure of the state equations is preserved while mathematical convenience is gained e.g., in diagonalized form each equation is decoupled from the rest facilitates analysis, e.g. modal analysis proportionality between the rate and state vectors is preserved Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 10 DOES THIS APPLY TO THE NONLINEAR CASE? The Hamiltonian form permits an analogous approach using canonical transformations. Any change of variables that preserves the value of the Hamiltonian preserves the structure of the Hamiltons equations is a canonical transformation. Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 11 EXAMPLE: The old displacements may be functions of both the new displacements and/or the new momenta. A simple canonical transformation p = q* q = p* old equations (conservative terms only) dp/dt = H(p,q)/q dq/dt = H(p,q)/p define H*(p*,q*) = H(p,q) the value of the Hamiltonian is preserved H*/p* = (H/q) (q/p*) = H/q H*/q* = (H/p) (p/q*) = H/p time differentiate the new coordinates dq*/dt = dp/dt = H/q = H*/p* dp*/dt = dq/dt = H(p,q)/p = H*/q* new equations (conservative terms only) dq*/dt = H*/p* dp*/dt = H*/q* the structure of Hamiltons equations is preserved Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 12 new equations with non-conservative terms dp*/dt = H(p*,q*)/q* + e*(p*,q*,t) dq*/dt = H(p*,q*)/p* f*(p*,q*,t) The forcing terms must be transformed as follows fk* = (ejpj/pk* + ejqj/pk*) j ek* = (fjpj/qk* + ejqj/qk*) j NOTE: Displacement and momentum may be exchanged canonical transformation may destroy the physical meaning of variables Parallel to transformation of a linear system to decoupled form the original real-valued physical variables become complex valued Mod. Sim. Dyn. Sys. Lagrangian and Hamiltonian forms page 13
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

MIT - MECHANICAL - 2.141
HAMILTON-JACOBI THEORY GOAL: Find a particular canonical transformation such that the &quot;new&quot; Hamiltonian is a function only of the &quot;new&quot; momenta. MATHEMATICAL PRELIMINARIES A canonical transformation may be derived from a generating function. Arguments
MIT - MECHANICAL - 2.141
HEAT TRANSFER AND THE SECOND LAW Thus far we've used the first law of thermodynamics: Energy is conserved. Where does the second law come in? One way is when heat flows. Heat flows in response to a temperature gradient. If two points are in thermal contac
MIT - MECHANICAL - 2.141
Parameterization, Analysis &amp; Simulation of a Heat GunSubmitted byThomas A. Bowers December 10, 20022.141: Modeling and Simulation of Dynamic Systems Fall 2002 Massachusetts Institute of Technology1 IntroductionThis paper discusses the dynamic analysi
MIT - MECHANICAL - 2.141
EXAMPLE: IDEAL GAS MANY LOW-DENSITY GASES AT MODERATE PRESSURES MAY BE ADEQUATELY MODELED AS IDEAL GASES. Are the ideal gas model equations compatible with models of dynamics in other domains? AN IDEAL GAS IS OFTEN CHARACTERIZED BY THE RELATION PV = mRT P
MIT - MECHANICAL - 2.141
Interaction Control Manipulation requires interaction object behavior affects control of force and motionIndependent control of force and motion is not possible object behavior relates force and motion contact a rigid surface: kinematic constraint mov
MIT - MECHANICAL - 2.141
Contact instability Problem: Contact and interaction with objects couples their dynamics into the manipulator control system This change may cause instability Example: integral-action motion controller coupling to more mass evokes instability Impedanc
MIT - MECHANICAL - 2.141
Kinematic transformation of mechanical behaviorNeville HoganGeneralized coordinates are fundamentalIf we assume that a linkage may accurately be described as a collection of linked rigid bodies, their generalized coordinates are a fundamental requireme
MIT - MECHANICAL - 2.141
LAGRANGE'S EQUATIONS (CONTINUED)Mechanism in &quot;uncoupled&quot; inertial coordinates: (innermost box in the figure)F = dp dt ; p = MvMechanism in generalized coordinates: (middle box in the figure) = d/dt Ek*/; = I(); Ek*(,) = tI()d L L * = with L(,) = Ek (,
MIT - MECHANICAL - 2.141
INERTIAL MECHANICS Neville Hogan The inertial behavior of a mechanism is substantially more complicated than that of a translating rigid body. Strictly speaking, the dynamics are simple; the underlying mechanical physics is still described by Newton's law
MIT - MECHANICAL - 2.141
EXAMPLE: THERMAL DAMPINGwork in air sealed outletA BICYCLE PUMP WITH THE OUTLET SEALED. When the piston is depressed, a fixed mass of air is compressed. -mechanical work is done. The mechanical work done on the air is converted to heat. -the air tempera
MIT - MECHANICAL - 2.141
NONLINEAR MECHANICAL SYSTEMS (MECHANISMS) The analogy between dynamic behavior in different energy domains can be useful. Closer inspection reveals that the analogy is not complete. One key distinction of mechanical systems is the role of kinematics - the
MIT - MECHANICAL - 2.141
ENERGY-STORING COUPLING BETWEEN DOMAINS MULTI-PORT ENERGY STORAGE ELEMENTS Context: examine limitations of some basic model elements. EXAMPLE: open fluid container with deformable walls P=gh h=AV V = Cf P where Cf = A g-fluid capacitor But when squeezed,
MIT - MECHANICAL - 2.141
NODICITY One of the important ways that physical system behavior differs between domains is the way elements may be connected. Electric circuit elements may be connected in series or in parallel - networks of arbitrary structure may be assembled This no
MIT - MECHANICAL - 2.141
Convection bonds and &quot;pseudo&quot; bondsEven in the simplest case of matter transport, power has two components, one due to the rate of work done, the other due to transported internal energy of the material. &quot;Pseudo&quot; bond graphs depict two distinct bonds. On
MIT - MECHANICAL - 2.141
MULTI-DOMAIN MODELING WHAT'S THE ISSUE? Why not just &quot;write down the equations&quot;? - standard formulations in different domains are often incompatible usually due to incompatible boundary conditions (choice of &quot;inputs&quot;) EXAMPLE: SIMPLE FLUID SYSTEM Scenario
MIT - MECHANICAL - 2.141
CAUSAL ANALYSISThings should be made as simple as possible - but no simpler. Albert Einstein How simple is &quot;as simple as possible&quot;? Causal assignment provides considerable insight.EXAMPLE: AQUARIUM AIR PUMPoscillatory motion in this direction coil leve
MIT - MECHANICAL - 2.141
EXAMPLE: ELECTROMAGNETIC SOLENOID A common electromechanical actuator for linear (translational) motion is a solenoid.Current in the coil sets up a magnetic field that tends to center the movable armature.Electromagnetic Solenoidpage 1 Neville HoganO
MIT - MECHANICAL - 2.141
CO-ENERGY (AGAIN) In the linear case, energy and co-energy are numerically equal. -the value of distinguishing between them may not be obvious. Why bother with co-energy at all? EXAMPLE: SOLENOID WITH MAGNETIC SATURATION. Previous solenoid constitutive eq
MIT - MECHANICAL - 2.141
LINEARIZED ENERGY-STORING TRANSDUCER MODELS Energy transduction in an electro-mechanical solenoid may be modeled by an energy-storing multiport.e= i..ICF . xEnergy transduction in an electric motor may be modeled by a gyrator.e= iGYF . xBut the
MIT - MECHANICAL - 2.141
Massachusetts Institute of Technology Department of Mechanical Engineering2.141 Modeling and Simulation of Dynamic SystemsINTRODUCTIONGOAL OF THE SUBJECT Methods for mathematical modeling of engineering systems Computational approaches are ubiquitous i
MIT - MECHANICAL - 2.141
Junction elements in network models. Classify by number of ports and examine the possible structures that result. Using only one-port elements, no more than two elements can be assembled.Combining two two-ports yields another two-port.At most two one-po
MIT - MECHANICAL - 2.141
EXAMPLE: THERMAL DAMPINGwork in air sealed outletA BICYCLE PUMP WITH THE OUTLET SEALED. When the piston is depressed, a fixed mass of air is compressed. -mechanical work is done. The mechanical work done on the air is converted to heat. -the air tempera
MIT - MECHANICAL - 2.141
NONLINEAR MECHANICAL SYSTEMS CANONICAL TRANSFORMATION S AND NUMERICAL INTEGRATION Jacobi Canonical Transformations A Jacobi canonical transformations yields a Hamiltonian that depends on only one of the conjugate variable sets. Assume dependence on new mo
MIT - MECHANICAL - 2.141
NETWORK MODELS OF TRANSMISSION LINES AND WAVE BEHAVIOR MOTIVATION: Ideal junction elements are power-continuous. Power out = power out instantaneously In reality, power transmission takes finite time. Power out power in Consider a lossless, continuous uni
MIT - MECHANICAL - 2.141
WORK-TO-HEAT TRANSDUCTION IN THERMO-FLUID SYSTEMS ENERGY-BASED MODELING IS BUILT ON THERMODYNAMICS - the fundamental science of physical processes. THERMODYNAMICS IS TO PHYSICAL SYSTEM DYNAMICS WHAT GEOMETRY IS TO MECHANICS. WHY SHOULD WE CARE ABOUT THERM
MIT - MECHANICAL - 2.154
MANEUVERING AND CONTROL OF MARINE VEHICLESMichael S. Triantafyllou Franz S. HoverDepartment of Ocean Engineering Massachusetts Institute of Technology Cambridge, Massachusetts USAManeuvering and Control of Marine Vehicles Latest Revision: November 5, 2
MIT - MECHANICAL - 2.154
111.1KINEMATICS OF MOVING FRAMESRotation of Reference FramesWe denote through a subscript the specific reference system of a vector. Let a vector ex pressed in the inertial frame be denoted as , and in a body-reference frame b . For the x x moment, w
MIT - MECHANICAL - 2.154
2VESSEL INERTIAL DYNAMICSWe consider the rigid body dynamics with a coordinate system affixed on the body. A common frame for ships, submarines, and other marine vehicles has the body-referenced xaxis forward, y-axis to port (left), and z-axis up. This
MIT - MECHANICAL - 2.154
3NONLINEAR COEFFICIENTS IN DETAILThe method of hydrodynamic coefficients is a somewhat blind series expansion of the fluid force in an attempt to provide a framework on which to base experiments and calculations to evaluate these terms. The basic dificu
MIT - MECHANICAL - 2.154
44.1VESSEL DYNAMICS: LINEAR CASESurface Vessel Linear ModelWe rst discuss some of the hydrodynamic parameters which govern a ship maneuvering in the horizontal plane. The b ody x-axis is forward and the y -axis is to port, so positive r has the vessel
MIT - MECHANICAL - 2.154
55.1SIMILITUDEUse of Nondimensional GroupsFor a consistent description of physical processes, we require that all terms in an equation must have the same units. On the basis of physical laws, some quantities are dependent on other, independent quantit
MIT - MECHANICAL - 2.154
6CAPTIVE MEASUREMENTSBefore making the decision to measure hydrodynamic derivatives, a preliminary search of the literature may turn up useful estimates. For example, test results for many hull-forms have already been published, and the basic lifting su
MIT - MECHANICAL - 2.154
7STANDARD MANEUVERING TESTSThis section describes some of the typical maneuvering tests which are performed on full-scale vessels, to assess stability and performance.7.1Dieudonn Spiral e1. Achieve steady speed and direction for one minute. No change
MIT - MECHANICAL - 2.154
88.1STREAMLINED BODIESNominal Drag ForceA symmetric streamlined body at zero angle of attack experiences only a drag force, which has the form 1 (109) FA = - CA Ao U 2 . 2 The drag coefficient CA has both pressure and skin friction components, and hen
MIT - MECHANICAL - 2.154
99.1SLENDER-BODY THEORYIntroductionConsider a slender body with d &lt; L, that is mostly straight. The body could be asymmetric in cross-section, or even flexible, but we require that the lateral variations are small and(Continued on next page)409 SLE
MIT - MECHANICAL - 2.154
1010.1PRACTICAL LIFT CALCULATIONSCharacteristics of Lift-Producing MechanismsAt a small angle of attack, a slender body experiences transverse force due to: helical body vortices, the blunt trailing end, and fins. The helical body vortices are stable
MIT - MECHANICAL - 2.154
11FINS AND LIFTING SURFACESVessels traveling at significant speed typically use rudders, elevators, and other streamlined control surfaces to maneuver. Their utility arises mainly from the high lift forces they can develop, with little drag penalty. Lif
MIT - MECHANICAL - 2.154
1212.1PROPELLERS AND PROPULSIONIntroductionWe discuss in this section the nature of steady and unsteady propulsion. In many marine vessels and vehicles, an engine (diesel or gas turbine, say) or an electric motor drives the(Continued on next page)12
MIT - MECHANICAL - 2.154
13ELECTRIC MOTORSModern underwater vehicles and surface vessels are making increased use of electrical ac tuators, for all range of tasks including weaponry, control surfaces, and main propulsion. This section gives a very brief introduction to the most
MIT - MECHANICAL - 2.154
14TOWING OF VEHICLESVehicles which are towed have some similarities to the vehicles that have been discussed so far. For example, towed vehicles are often streamlined, and usually need good directional stability. Some towed vehicles might have active li
MIT - MECHANICAL - 2.154
15TRANSFER FUNCTIONS &amp; STABILITYThe reader is referred to Laplace Transforms in the section MATH FACTS for preliminary material on the Laplace transform. Partial fractions are presented here, in the context of control systems, as the fundamental link be
MIT - MECHANICAL - 2.154
1616.116.1.1CONTROL FUNDAMENTALSIntroductionPlants, Inputs, and OutputsController design is about creating dynamic systems that behave in useful ways. Many target systems are physical; we employ controllers to steer ships, fly jets, position electri
MIT - MECHANICAL - 2.154
1717.1MODAL ANALYSISIntroductionThe evolution of states in a linear system occurs through independent modes, which can be driven by external inputs, and observed through plant output. This section provides the basis for modal analysis of systems. Thro
MIT - MECHANICAL - 2.154
(Continued on next page)18.2 Roots of Stability Nyquist Criterion87S(s) =e(s) 1 = , r(s) 1 + P (s)C(s)where P (s) represents the plant transfer function, and C(s) the compensator. The closedloop characteristic equation, whose roots are the poles of t
MIT - MECHANICAL - 2.154
1919.1LINEAR QUADRATIC REGULATORIntroductionThe simple form of loopshaping in scalar systems does not extend directly to multivariable (MIMO) plants, which are characterized by transfer matrices instead of transfer functions. The notion of optimality
MIT - MECHANICAL - 2.154
2020.1KALMAN FILTERIntroductionIn the previous section, we derived the linear quadratic regulator as an optimal solution for the full-state feedback control problem. The inherent assumption was that each state was known perfectly. In real applications
MIT - MECHANICAL - 2.154
2121.1LOOP TRANSFER RECOVERYIntroductionThe Linear Quadratic Regulator(LQR) and Kalman Filter (KF) provide practical solutions to the full-state feedback and state estimation problems, respectively. If the sensor noise and disturbance properties of th
MIT - MECHANICAL - 2.154
2222.122.1.1APPENDIX 1: MATH FACTSVectorsDefinitionA vector has a dual definition: It is a segment of a a line with direction, or it consists of its projection on a reference system 0xyz, usually orthogonal and right handed. The first form is indepe
MIT - MECHANICAL - 2.154
23 APPENDIX 2: ADDED MASS VIA LAGRANGIAN DYNAMICSThe development of rigid body inertial dynamics presented in a previous section depends on the rates of change of vectors expressed in a moving frame, specifically that of the vehicle. An alternative appro
MIT - MECHANICAL - 2.154
24APPENDIX 3: LQR VIA DYNAMIC PROGRAM MINGThere are at least two conventional derivations for the LQR; we present here one based on dynamic programming, due to R. Bellman. The key observation is best given through a loose example.(Continued on next pag
MIT - MECHANICAL - 2.154
25Further Robustness of the LQRThe most common robustness measures attributed to the LQR are a one-half gain reduction in any input channel, an infinite gain amplification in any input channel, or a phase error of plus or minus sixty degrees in any inpu
MIT - MECHANICAL - 2.160
Department of Mechanical Engineering Massachusetts Institute of Technology2.160 Identification, Estimation, and Learning End-of-Term ExaminationMay 17, 2006 1:00 3:00 pm (12:30 2:30 pm) Close book. Two sheets of notes are allowed. Show how you arrived a
MIT - MECHANICAL - 2.160
Identification, Estimation, and Learning3-0-9 H-Level Graduate Credit Prerequisite: 2.151 or similar subject2.160Reference Books BooksLennart Ljung, &quot;System Identification: Theory for the User, Second Edition&quot;, Prentice-Hall 1999 Graham Goodwin and
MIT - MECHANICAL - 2.160
2.160 Identification, Estimation, and Learning Lecture Notes No. 1February 8, 2006Mathematical models of real-world systems are often too difficult to build based on first principles alone.Figure by MIT OCW. Figure by MIT OCW.System Identification; &quot;L
MIT - MECHANICAL - 2.160
2.160 Identification, Estimation, and LearningLecture Notes No. 2February 13, 2006 2. Parameter Estimation for Deterministic Systems 2.1 Least Squares Estimationu1 u2 M umMDeterministic System w/parameter Linearly parameterized modelyInput-output
MIT - MECHANICAL - 2.160
2.160 Identification, Estimation, and LearningLecture Notes No. 3February 15, 2006 2.3 Physical Meaning of Matrix PThe Recursive Least Squares (RLS) algorithm updates the parameter vector ^(t - 1) based on new data T (t ), y (t ) in such a way that the
MIT - MECHANICAL - 2.160
2.160 Identification, Estimation, and LearningLecture Notes No. 4February 17, 2006 3. Random Variables and Random ProcessesDeterministic System: Input OutputIn realty, the observed output is noisy and does not fit the model perfectly. In the determini
MIT - MECHANICAL - 2.160
2.160 System Identification, Estimation, and LearningLecture Notes No. 5February 22, 2006 4. Kalman Filtering4.1 State Estimation Using ObserversIn discrete-time form a linear time-varying, deterministic, dynamical system is represented by xt +1 = At
MIT - MECHANICAL - 2.160
2.160 System Identification, Estimation, and LearningLecture Notes No. 6February 24, 2006 4.5.1 The Kalman Gain Consider the error of a posteriori estimate xt et xt xt = xt t 1 + K (yt Ht xt t 1 )xt t = xt t 1 +K t (Ht xt +vt H t xt t 1
MIT - MECHANICAL - 2.160
2.160 System Identification, Estimation, and LearningLecture Notes No. 7March 1, 20064.7. Continuous Kalman FilterConverting the Discrete Filter to a Continuous FilterContinuous process x = Fx + Gw(t )(49) (50)Measurement Assumptionsy = Hx + v(t
MIT - MECHANICAL - 2.160
2.160 System Identification, Estimation, and LearningLecture Notes No. 8March 6, 20064.9 Extended Kalman FilterIn many practical problems, the process dynamics are nonlinear.w Process DynamicsvyuKalman Gain &amp; Covariance Update+ _Model (Lineariz