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9 Pages

### asymmetric_junct

Course: MECHANICAL 2.141, Fall 2006
School: MIT
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Word Count: 1126

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asymmetric Ideal junction elements Relax the symmetry assumption and examine the resulting junction structure. For simplicity, consider two-port junction elements. As before, assume instantaneous power transmission between the ports without storage or dissipation of energy. Characterize the power flow in and out of a twoport junction structure using four real-valued wave-scattering variables. Using vector...

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asymmetric Ideal junction elements Relax the symmetry assumption and examine the resulting junction structure. For simplicity, consider two-port junction elements. As before, assume instantaneous power transmission between the ports without storage or dissipation of energy. Characterize the power flow in and out of a twoport junction structure using four real-valued wave-scattering variables. Using vector notation: u1 u = u 2 v1 v = v 2 (A.1) (A.2) The input and output power flows are the square of the length of these vectors, their inner products. Pin = ui2 = utu i=1 2 (A.3) Pout = vi2 = vtv i=1 2 (A.4) The constitutive equations of the junction structure may be written as follows. v = f(u) (A.5) Geometrically, the requirement that power in equal power out means that the length of the vector v must equal the length of the vector u, i.e. their tips must lie on the perimeter of a circle (see figure A.1). For any two particular values of u and v, the algebraic relation f(.) is equivalent to a rotation operator. v = S(u) u where the square matrix S is known as a scattering matrix. (A.6) Mod. Sim. Dyn. Sys. page 1 Neville Hogan u2 v2 u u1 v1 v S need not be a constant matrix, but may in general depend on the power flux through the junction, hence the notation S(u). However, S is subject to important restrictions. In particular, vtv = utStSu = utu (A.7) S is an orthogonal matrix: the vectors formed by each of its rows (or columns) are (i) orthogonal and (ii) have unit magnitude; its transpose is its inverse. S tS = 1 This constrains the coefficients of the scattering matrix as follows. a b S=c d (A.8) (A.9) (A.10) (A.11) (A.12) page 2 Neville Hogan a2 + c2 = 1 ab + cd = 0 b2 + d2 = 1 Mod. Sim. Dyn. Sys. As there are only three independent equations and four unknown quantities, we see that this junction is characterized by a single parameter. We may also write the orthogonality condition as SSt = 1 which yields the following equations. a2 + b2 = 1 ac + bd = 0 c2 + d2 = 1 (A.14) (A.15) (A.16) (A.13) There are four possible solutions to these equations. Combining A.10 and A.16, a2 = 1 - c2 = d2. Thus a = d. If a = d then b =c = 1 - a2 . One solution Choosing the positive root yields one solution. Assuming the coefficient a to be the undetermined parameter, a S= - 1 - a2 1 - a2 a (A.17) Rewrite in terms of effort and flow variables. e1 e = e 2 f1 f = f 2 (A.18) (A.19) The relation between efforts and wave-scattering variables is as follows. e = (u - v) c = c (1 - S) u (A.20) where c is a scaling constant. The relation between flows and wave-scattering variables is as follows. Mod. Sim. Dyn. Sys. page 3 Neville Hogan f = (u + v)/c = 1/c (1 + S) u (A.21) If |a| 1 then 1 + S and 1 - S are nonsingular matrices and the input wave scattering variables u1 and u2 may be eliminated as follows. e = c2 (1 - S) (1 + S)-1 f e1 2 =c e2 (A.22) - (1 - a)/(1 + a) f1 0 f2 0 (1 - a)/(1 + a) (A.23) Writing G = c2 (1 - a)/(1 + a) we obtain the equation for an ideal gyrator. e1 0 -G f1 = G 0 f2 e2 (A.24) Note that equations A.20 and A.21 imply a sign convention in effort-flow coordinates such that power is positive inwards on both ports. Pnet inwards = etf = utu - vtv (A.25) To follow the more common sign convention we simply may change the sign of f2 in equation A.24. If a = 1, e is identically zero for all values of f. No energy is exchanged between the ports and the junction structure behaves like a dissipator with zero resistance. If a = -1, f is identically zero for all values of e. No energy is exchanged between the ports and the junction structure behaves like a dissipator with infinite resistance (zero conductance). Mod. Sim. Dyn. Sys. page 4 Neville Hogan A second solution Choosing a = d and using the negative root yields another solution. Again assuming the coefficient a to be the undetermined parameter, S= a 1 - a2 - 1 - a2 a (A.26) In this case the relation between efforts and flows is e1 0 2 =c - (1 - a)/(1 + a) e2 (1 - a)/(1 + a) f1 0 f2 (A.27) Again we obtain the equation for an ideal gyrator. e1 0 G f1 = -G 0 f2 e2 (A.28) Mod. Sim. Dyn. Sys. page 5 Neville Hogan A third solution If a = - d, b = c = 1 - a2 . Using the positive root and assuming a to be the undetermined parameter S= a 1a2 1 - a2 -a (A.29) In this case the matrices 1 + S and 1 - S are singular for all values of the parameter a. However, equations A.20 and A.21 may be combined as follows: 1 1 - S e2/c = ----- u1 cf1 1 + Su2 cf2 1 1-a e2/c - 1 - a2 cf = 1 + a 1 cf2 1 - a2 e2/c - 1 - a2 = 1+a cf1 e1/c = cf2 e /c (A.30) e /c - 1 - a2 1+a 1 - a2 1-a u1 u2 (A.31) If |a| 1, the 4 x 2 matrix relating efforts and flows to the input scattering variables contains two nonsingular 2 x 2 submatrices. 1 + a u1 1 - a2 u2 u2 (A.32) 1-a 1a2 - 1 - a2 u1 1-a (A.33) Solving the second of these for u and substituting into the first we obtain a relation between efforts and flows. Mod. Sim. Dyn. Sys. page 6 Neville Hogan e2 - (1 + a)/(1 - a) = 0 f1 e1 (1 + a)/(1 - a) f2 0 (A.34) Writing T = (1 + a)/(1 - a) we obtain the equation for an ideal transformer. e2 -T 0 e1 = 0 T f2 f1 (A.35) To follow the more common sign convention we may change the sign of e2. If the parameter a = 1, an argument similar to that used above shows that a degenerate case results in which no energy is exchanged between the ports. Mod. Sim. Dyn. Sys. page 7 Neville Hogan Final solution Choosing a = d and using the negative root we obtain the fourth solution. a - 1 - a2 S= -a - 1 - a2 (A.36) Once again, the matrices 1 + S and 1 - S are singular for all values of the parameter a, but by rearranging equations A.20 and A.21 as before the corresponding relation between efforts and flows is e2 = f1 (1 + a)/(1 - a) 0 e1 - (1 + a)/(1 - a) f2 0 (A.37) Again we obtain the equation for an ideal transformer e2 T 0 e1 = 0 -T f2 f1 (A.38) Mod. Sim. Dyn. Sys. page 8 Neville Hogan Two-port junction elements There are only two possible power-continuous, asymmetric two-port junction elements, the gyrator and the transformer. Unlike the ideal symmetric junction elements (0 and 1) the ideal asymmetric junction elements may be nonlinear. The relation between efforts and flows must have a multiplicative form. The general asymmetric junction elements are a modulated gyrator (MGY) and a modulated transformer (MTF) respectively. Mod. Sim. Dyn. Sys. page 9 Neville Hogan
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MIT - MECHANICAL - 2.141
NETWORK MODELS OF BERNOULLI'S EQUATION The phenomenon described by Bernoulli's equation arises from momentum transport due to mass flow. EXAMPLE: A PIPE OF VARYING CROSS-SECTION.section 1 Q1 section 2 A2 Q2 v2 P2A 1 v1 P1 Assume: incompressible flow
MIT - MECHANICAL - 2.141
AMPLIFIERS A circuit containing only capacitors, amplifiers (transistors) and resistors may resonate. A circuit containing only capacitors and resistors may not. Why does amplification permit resonance in a circuit with only one kind of storage element?A
MIT - MECHANICAL - 2.141
BLOCK DIAGRAMS, BOND GRAPHS AND CAUSALITY The main purpose of modeling is to develop insight. &quot;Drawing a picture&quot; of a model promotes insight. Why not stick with the familiar block diagrams? Block diagrams provide a picture of equations; -they portray ope
MIT - MECHANICAL - 2.141
T h e Basic Bond Graph Primitives Fundamental quantities and relations P : p o w e r e: effort p: m o m e n t u m e=f: flow dpldt dq/dt DenotesE : energyq: displacement f Bond Graph Symbol=Electrical Network IconTypical M echanical Iconpower p
MIT - MECHANICAL - 2.141
Stirling EngineMarten Byl 12/12/021xTe R Th Tc=0Figure 1: Schematic of Stirling Engine with key variables noted.IntroductionIn the undergraduate class 2.670 at M.I.T., the students explore basic manufacturing tech niques by building a stirling eng
MIT - MECHANICAL - 2.141
REVIEW NETWORK MODELING OF PHYSICAL SYSTEMSa.k.a. &quot;lumped-parameter&quot; modelingEXAMPLE: VIBRATION IN A CABLE HOIST Problem The cage of an elevator is hoisted by a long cable wound over a drum driven through a gear-set by an electric motor. The motor is re
MIT - MECHANICAL - 2.141
REVIEW NETWORK MODELING OF PHYSICAL SYSTEMS EXAMPLE: VIBRATION IN A CABLE HOIST Bond graphs of the cable hoist models help to develop insight about how the electrical R-C filter affects the mechanical system dynamics. Equivalent mechanical system: velocit
MIT - MECHANICAL - 2.141
CANONICAL TRANSFORMATION THEORY A canonical transformation may express new displacements and momenta as functions of both the original displacements and momenta, but is restricted such that it preserves the Hamiltonian form of the differential equations.
MIT - MECHANICAL - 2.141
Capstan-a mechanical amplifierPhotograph removed due to copyright restrictions. rFFnormalv control Fcontrol v out FoutF + FA schematic diagram of a basic capstan and a force diagram for a small segment of the rope are shown in the figures. Fnormal
MIT - MECHANICAL - 2.141
CONVECTION AND MATTER TRANSPORT PROCESSES REVIEW: CLOSED SYSTEM Simple substance i.e., no reacting components internal energy U = U(S,V,m) constant mass makes this a two-port capacitor - one port for each variable argument of the energy function displacem
MIT - MECHANICAL - 2.141
MATTER TRANSPORT (CONTINUED) There seem to be two ways to identify the effort variable for mass flow gradient of the energy function with respect to mass is &quot;matter potential&quot;, - (molar) specific Gibbs free energy power dual of mass flow appears to be (m
MIT - MECHANICAL - 2.141
Magnetic electro-mechanical machinesNeville Hogan This is a brief outline of the physics underlying simple electro-magnetic machines, especially the ubiquitous direct-current permanent-magnet motor.Lorentz ForceA magnetic field exerts force on a moving
MIT - MECHANICAL - 2.141
ENTROPY PRODUCTION AND NONLINEARITY. Is entropy production an exclusively nonlinear phenomenon? Must it always vanish in a linearized model? Consider simple heat transfer modeled by Fourier's law: Q = (kA/l)(T1 - T2) where Q is heat flow rate, k is therma
MIT - MECHANICAL - 2.141
NONLINEAR MECHANICAL 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 t
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