Electromagnetics, Control and Robotics: A Problems & Solutions Approach

622 Pages·2022·18.439 MB·English

by  Harish Parthasarathy

Electromagnetics, Control and Robotics
A Problems & Solutions Approach

Electromagnetics, Control and Robotics
A Problems & Solutions Approach
Harish Parthasarathy
Professor
Electronics & Communication Engineering
Netaji Subhas Institute of Technology (NSIT)
Harish Parthasarathy
New Delhi, Delhi-110078
Firstpublished2023
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Preface
This book is a research monograph covering a variety of problems in statistical
state and parameter estimation in nonlinear stochastic dynamical system in
both the classical and quantum scenario, propagation of electromagnetic
waves in a plasma described by the Boltzmann Kinetic Transport Equation,
Classical and Quantum General Relativity etc. It will be of use to Engineering
undergraduate students interested in analysing the motion of robots subject to
random perturbation. It will also be very useful for research scientists working
in Quantum Filtering and studying problems of inhomogeneous perturbation
to in a homogeneous and isotropic universe leading to the formation of galaxies
in an expanding universe.
The mathematical prerequisite for studying this book include Ordinary and
Partial Differential Equations, Riemannian Geometry, Stochastic Calculus of
Brownian Motion and Poisson Processes, Perturbation Theory of Dynamical
System, Group Theory and Group Representation Theory and Basic Operator
Theory Especially in the Boson Fock Space as formulated by R.L. Hudson &
K.R. Parthasarathy.
Author
v

Table of Contents
1. Master-slave Robots, Optimal Control, Quantum
Mechanics and Information 1–54
2. Quantum State Estimation Robot System Estimation,
Rotation Matrix Estimation, Quantum Filtering 55–100
3. Quantum Field Theory, Gravitational and Electromagnetic
Waves in Cosmology, Electromagnetics and Fluids,
Quantum Gate Design 101–134
4. Brownian Motion and Levy Processes, Nonlinear Filtering
Plasma Waves, Quantum Stochastics and Parameter Estimation 135–186
5. Quantum Gravity, Lie Groups in Robotics, Field of Robots,
Quantum Robots, Quantum Transmission Lines,
Quantum Optimal Control 187–216
6. Random Feedback Control, Group Invariants and Pattern
Classification, Quantum Mechanics in Levy Noise,
State Observer and Trajectory Tracking 217–250
7. Radon Transform for Rotation Estimation, Interaction
Picture in Quantum Mechanics, Curvature Tensor in GTR,
Gauge Fields 251–294
8. Spin-field Interaction, Viscous and Thermal Effects in GTR,
Channel Capacity, Filtering and Control for Robots,
Discrete Time Nonlinear Stochastic Filter 295–366
9. Quantum Filtering in Robotics, Information, Feynman
Path Integrals, Levy Noise, Haar Measure on Groups,
Gravity and Robots, Canonical Quantum Gravity, Langevin
Equation, Antenna Current in a Field Dependent Medium 367–428
10. Perturbation Theory in Cosmology, Hamiltonian of Charged
Particles in Curved Space-time, Brownian Local Time, Statistics
of Quantum Observables,Quantization of Stochastic Dynamical
Systems, Quantum Relative Entropy Evolution for Open Systems,
Quantum Gates in Levy Noise 429–506
11. Wave Equation with Random Boundary, Robots in a Fluid,
Cosmology with Stochastic Perturbations, Interaction Between
Graviton, Photon and Electron- Positron Fields, Evans-Hudson
Flows for Quantization of Stochastic Evolutions 507–608
12. Appendix 609–610
13. References 611–613
vii

Master-slave Robots, Optimal Control,
Quantum Mechanics and Information
1
[1] Basic Markov chain theory:
Let X (t), t ≥ be a stochastic process on a probability space taking values in Z with
the property
P (X (t) ∈ B | X (τ) : τ ≤ s) = P (X (t) ∈ B | X(s))
for all B ⊂ Z and 0 ≤ s ≤ t. Then X (t) is called a Markov chain. It is completely
characterized by the infinitesimal generator L = ((l )) where
ab
P (X (t + dt) = b| X(t) = a) = d + dt. l + 0 (dt)
a, b ab
Let P (t) = P (X(t) = a)
a
for some initial probability distribution of X (0) on Z. Then, the Chapman -
Kolmogorov equation
P′(t) = ∑P (t)λ
a b ba
b
or equivalently, in terms of matrices.
P′(t)T = P(t)T Λ
which gives
P(t)T = P(0)T exp (tL)
Let 0 ≤ t1 < t2< ...< tp. Then for f : Z→ R, we have
E(f(X(t1)...f(X(tp)) = P(0)T exp(t1L)Df..
exp((t – t )L)D...exp((t – t )L)Dfu
2 1 f p p–1
where
Df = diag(f(a) : a ∈ Z)
and
u = [1,1,...]T
Let T <T <...be the jump times of the Markov chain. Assume that X(0)=i . Then
1 2 0
for i ≠i ,
1 0 P(T (t, t + dt], X(T +) = i P(T > t) l(i i )dt
1 1 1 1 0,1
Now,
P(T >t + dt) = P(T > t) (1+l(i , i ) dt)
1 1 0 0
and hence
P(T >t) = exp(tl(i i )
1 0, 0
We are using the notation l(a, b) in place of l . Thus, we get the joint distribution
ab
of (T , X(T +)):
1 1
P(T ∈(t, t + dt),X(T +) = i ) = l(i , i ).exp(tl(i , i ))dt
1 1 1 0 1 0 0
Likewise, we get using the Markov property
P(T ∈(t,t + dt), X (T +) = i |T ,(X (T +) = i )
n n n n – 1, n–1 n – 1
1