Second part of Matrices at undergraduate in science (math, physics, engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
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http://www.solohermelin.com.
Estimate the hidden States of a Non-linear Dynamic Stochastic System from Noisy Measurements. Estimation is a prerequisite. The Probability Theory summary is included.
The presentation is at graduate level in math and engineering.
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Mathematical description of Legendre Functions.
Presentation at Undergraduate in Science (math, physics, engineering) level.
Please send any comments or suggestions to improve to solo.hermelin@gmail.com.
More presentations can be found on my website at http://www.solohermelin.com.
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
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For more presentations on different subjects visit my website at http://www.solohermelin.com.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
Second part of Matrices at undergraduate in science (math, physics, engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com.
For more presentations visit my website at
http://www.solohermelin.com.
Estimate the hidden States of a Non-linear Dynamic Stochastic System from Noisy Measurements. Estimation is a prerequisite. The Probability Theory summary is included.
The presentation is at graduate level in math and engineering.
For comments please connect me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Mathematical description of Legendre Functions.
Presentation at Undergraduate in Science (math, physics, engineering) level.
Please send any comments or suggestions to improve to solo.hermelin@gmail.com.
More presentations can be found on my website at http://www.solohermelin.com.
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects please visit my homepage at http://www.solohermelin.com.
This presentation is in the Physics folder.
Equation of motion of a variable mass system3Solo Hermelin
This is the third of three presentations (self content) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. It uses the Lagrangian approach. It is recommended to see the first presentation before this one. Each presentation uses a different method of derivation..
This is the more difficult of the three presentations.
The presentation is at undergraduate (physics, engineering) level.
Please sent comments for improvements to solo.hermelin@gmail.com. Thanks!
For more presentations on different subjects please visit my website at http://www.solohermelin.com
Stochastic Processes describe the system derived by noise.
Level of graduate students in mathematics and engineering.
Probability Theory is a prerequisite.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
The Cramér-Rao Lower Bound on the Variance of the Estimator of continuous SISO, MIMO and Digital Systems.
Probability and Estimation are prerequisite.
For comments please connect me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
Dyadics algebra.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
4 matched filters and ambiguity functions for radar signals-2Solo Hermelin
Matched filters (Part 2of 2) maximizes the output signal-to-noise ratio for a known radar signal at a predefined time.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
Hello, I am Subhajit Pramanick. I and my friend, Sougata Dandapathak, both presented this ppt in our college seminar. It is basically based on the origin of calculus of variation. It consists of several topics like the history of it, the origin of it, who developed it, application of it, advantages and disadvantages etc. The main aim of this presentation is to increase our mathematical as well as physical conception on advanced classical mechanics. We hope you will all enjoy by reading this presentation. Thank you.
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
Equation of motion of a variable mass system1Solo Hermelin
This is the first of three presentations (the easiest one) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. Each presentation uses a different method of derivation.
The presentation is at undergraduate (physics, engineering) level.
Please sent comments for improvements to solo.hermelin@gmail.com. Thanks!
For more presentations on different subjects please visit my website at http://www.solohermelin.com.
Equation of motion of a variable mass system3Solo Hermelin
This is the third of three presentations (self content) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. It uses the Lagrangian approach. It is recommended to see the first presentation before this one. Each presentation uses a different method of derivation..
This is the more difficult of the three presentations.
The presentation is at undergraduate (physics, engineering) level.
Please sent comments for improvements to solo.hermelin@gmail.com. Thanks!
For more presentations on different subjects please visit my website at http://www.solohermelin.com
Stochastic Processes describe the system derived by noise.
Level of graduate students in mathematics and engineering.
Probability Theory is a prerequisite.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
The Cramér-Rao Lower Bound on the Variance of the Estimator of continuous SISO, MIMO and Digital Systems.
Probability and Estimation are prerequisite.
For comments please connect me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
Dyadics algebra.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Rotation in 3d Space: Euler Angles, Quaternions, Marix DescriptionsSolo Hermelin
Mathematics of rotation in 3d space, a lecture that I've prepared.
This presentation is at a Undergraduate in Science (Math, Physics, Engineering) level.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks!
Fore more presentations, please visit my website at
http://www.solohermelin.com/
4 matched filters and ambiguity functions for radar signals-2Solo Hermelin
Matched filters (Part 2of 2) maximizes the output signal-to-noise ratio for a known radar signal at a predefined time.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Presentation of the work on Prime Numbers.
intended for mathematics loving people.
Please send comments and suggestions for improvement to solo.hermelin@gmail.com.
More presentations can be found in my website at http://solohermelin.com.
Hello, I am Subhajit Pramanick. I and my friend, Sougata Dandapathak, both presented this ppt in our college seminar. It is basically based on the origin of calculus of variation. It consists of several topics like the history of it, the origin of it, who developed it, application of it, advantages and disadvantages etc. The main aim of this presentation is to increase our mathematical as well as physical conception on advanced classical mechanics. We hope you will all enjoy by reading this presentation. Thank you.
Describes the mathematics of the Calculus of Variations.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website on http://www.solohermelin.com
Equation of motion of a variable mass system1Solo Hermelin
This is the first of three presentations (the easiest one) for derivation of equations of motions of a variable mass system containing moving solids (rotors, pistons,..) and elastic parts. Each presentation uses a different method of derivation.
The presentation is at undergraduate (physics, engineering) level.
Please sent comments for improvements to solo.hermelin@gmail.com. Thanks!
For more presentations on different subjects please visit my website at http://www.solohermelin.com.
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...Wireilla
In this paper, the approximate controllability of impulsive linear fuzzy stochastic differential equations with nonlocal conditions in Banach space is studied. By using the Banach fixed point theorems, stochastic analysis, fuzzy process and fuzzy solution, some sufficient conditions are given for the approximate controllability of the system.
APPROXIMATE CONTROLLABILITY RESULTS FOR IMPULSIVE LINEAR FUZZY STOCHASTIC DIF...ijfls
In this paper, the approximate controllability of impulsive linear fuzzy stochastic differential equations with nonlocal conditions in Banach space is studied. By using the Banach fixed point
theorems, stochastic analysis, fuzzy process and fuzzy solution, some sufficient conditions are given for
the approximate controllability of the system.
MATLAB sessions: Laboratory 6
MAT 275 Laboratory 6
Forced Equations and Resonance
In this laboratory we take a deeper look at second-order nonhomogeneous equations. We will concentrate
on equations with a periodic harmonic forcing term. This will lead to a study of the phenomenon known
as resonance. The equation we consider has the form
d2y
dt2
+ c
dy
dt
+ ω20y = cosωt. (L6.1)
This equation models the movement of a mass-spring system similar to the one described in Laboratory
5. The forcing term on the right-hand side of (L6.1) models a vibration, with amplitude 1 and frequency
ω (in radians per second = 12π rotation per second =
60
2π rotations per minute, or RPM) of the plate
holding the mass-spring system. All physical constants are assumed to be positive.
Let ω1 =
√
ω20 − c2/4. When c < 2ω0 the general solution of (L6.1) is
y(t) = e−
1
2 ct(c1 cos(ω1t) + c2 sin(ω1t)) + C cos (ωt− α) (L6.2)
with
C =
1√
(ω20 − ω2)
2
+ c2ω2
, (L6.3)
α =
⎧
⎨
⎩
arctan
(
cω
ω20−ω2
)
if ω0 > ω
π + arctan
(
cω
ω20−ω2
)
if ω0 < ω
(L6.4)
and c1 and c2 determined by the initial conditions. The first term in (L6.2) represents the complementary
solution, that is, the general solution to the homogeneous equation (independent of ω), while the second
term represents a particular solution of the full ODE.
Note that when c > 0 the first term vanishes for large t due to the decreasing exponential factor.
The solution then settles into a (forced) oscillation with amplitude C given by (L6.3). The objectives of
this laboratory are then to understand
1. the effect of the forcing term on the behavior of the solution for different values of ω, in particular
on the amplitude of the solution.
2. the phenomena of resonance and beats in the absence of friction.
The Amplitude of Forced Oscillations
We assume here that ω0 = 2 and c = 1 are fixed. Initial conditions are set to 0. For each value of ω, the
amplitude C can be obtained numerically by taking half the difference between the highs and the lows
of the solution computed with a MATLAB ODE solver after a sufficiently large time, as follows: (note
that in the M-file below we set ω = 1.4).
1 function LAB06ex1
2 omega0 = 2; c = 1; omega = 1.4;
3 param = [omega0,c,omega];
4 t0 = 0; y0 = 0; v0 = 0; Y0 = [y0;v0]; tf = 50;
5 options = odeset(’AbsTol’,1e-10,’RelTol’,1e-10);
6 [t,Y] = ode45(@f,[t0,tf],Y0,options,param);
7 y = Y(:,1); v = Y(:,2);
8 figure(1)
9 plot(t,y,’b-’); ylabel(’y’); grid on;
c⃝2011 Stefania Tracogna, SoMSS, ASU 1
MATLAB sessions: Laboratory 6
10 t1 = 25; i = find(t>t1);
11 C = (max(Y(i,1))-min(Y(i,1)))/2;
12 disp([’computed amplitude of forced oscillation = ’ num2str(C)]);
13 Ctheory = 1/sqrt((omega0^2-omega^2)^2+(c*omega)^2);
14 disp([’theoretical amplitude = ’ num2str(Ctheory)]);
15 %----------------------------------------------------------------
16 function dYdt = f(t,Y,param)
17 y = Y(1); v = Y(2);
18 omega0 = param(1); c = param(2); omega = param(3);
19 dYdt = [ v ; cos(omega ...
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Some Common Fixed Point Theorems for Multivalued Mappings in Two Metric SpacesIJMER
In this paper we prove some common fixed point theorems for multivalued mappings in two
complete metric spaces.
AMS Mathematics Subject Classification: 47H10, 54H25
Similar to Inner outer and spectral factorizations (20)
Aircraft Susceptibility and Vulnerability.
This is from the last presentations from my side. Medical Problems prevent me to continue with new presentations.Please do not contact me.
Describes concepts and development of flying cars and other flying vehicles. Reference are given including to YouTube movies. At the end my view of Main Requirements and the related Design Requirements for a SkyCar are given. The main conclusion is that technologically we are ready to develop and product such a SkyCar in a few years.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Fighter Aircraft Performance, Part II of two, describes the parameters that affect aircraft performance.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Fighter Aircraft Performance, Part I of two, describes the parameters that affect aircraft performance.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Air Combat History describes the main air combats and fighter aircraft, from the beginning of aviation. The additional Youtube links are an important part of the presentation. A list of Air-to-Air Missile from different countries. is also given
For comments please contact me at solo.hermelin@gmail.com.
For more presentations visit my website at http://www.solohermelin.com.
Aerodynamics Part III of 3 describes aerodynamics of wings in supersonic flight.
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For more presentations on different subjects visit my website at http://www.solohermelin.com.
Aerodynamics Part II of 3 describes aerodynamics of bodies in supersonic flight.
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For more presentations on different subjects visit my website at http://www.solohermelin.com.
Aerodynamics Part I of 3 describes aerodynamics of wings and bodies in subsonic flight.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
The ability to recreate computational results with minimal effort and actionable metrics provides a solid foundation for scientific research and software development. When people can replicate an analysis at the touch of a button using open-source software, open data, and methods to assess and compare proposals, it significantly eases verification of results, engagement with a diverse range of contributors, and progress. However, we have yet to fully achieve this; there are still many sociotechnical frictions.
Inspired by David Donoho's vision, this talk aims to revisit the three crucial pillars of frictionless reproducibility (data sharing, code sharing, and competitive challenges) with the perspective of deep software variability.
Our observation is that multiple layers — hardware, operating systems, third-party libraries, software versions, input data, compile-time options, and parameters — are subject to variability that exacerbates frictions but is also essential for achieving robust, generalizable results and fostering innovation. I will first review the literature, providing evidence of how the complex variability interactions across these layers affect qualitative and quantitative software properties, thereby complicating the reproduction and replication of scientific studies in various fields.
I will then present some software engineering and AI techniques that can support the strategic exploration of variability spaces. These include the use of abstractions and models (e.g., feature models), sampling strategies (e.g., uniform, random), cost-effective measurements (e.g., incremental build of software configurations), and dimensionality reduction methods (e.g., transfer learning, feature selection, software debloating).
I will finally argue that deep variability is both the problem and solution of frictionless reproducibility, calling the software science community to develop new methods and tools to manage variability and foster reproducibility in software systems.
Exposé invité Journées Nationales du GDR GPL 2024
ANAMOLOUS SECONDARY GROWTH IN DICOT ROOTS.pptxRASHMI M G
Abnormal or anomalous secondary growth in plants. It defines secondary growth as an increase in plant girth due to vascular cambium or cork cambium. Anomalous secondary growth does not follow the normal pattern of a single vascular cambium producing xylem internally and phloem externally.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
DERIVATION OF MODIFIED BERNOULLI EQUATION WITH VISCOUS EFFECTS AND TERMINAL V...Wasswaderrick3
In this book, we use conservation of energy techniques on a fluid element to derive the Modified Bernoulli equation of flow with viscous or friction effects. We derive the general equation of flow/ velocity and then from this we derive the Pouiselle flow equation, the transition flow equation and the turbulent flow equation. In the situations where there are no viscous effects , the equation reduces to the Bernoulli equation. From experimental results, we are able to include other terms in the Bernoulli equation. We also look at cases where pressure gradients exist. We use the Modified Bernoulli equation to derive equations of flow rate for pipes of different cross sectional areas connected together. We also extend our techniques of energy conservation to a sphere falling in a viscous medium under the effect of gravity. We demonstrate Stokes equation of terminal velocity and turbulent flow equation. We look at a way of calculating the time taken for a body to fall in a viscous medium. We also look at the general equation of terminal velocity.
Seminar of U.V. Spectroscopy by SAMIR PANDASAMIR PANDA
Spectroscopy is a branch of science dealing the study of interaction of electromagnetic radiation with matter.
Ultraviolet-visible spectroscopy refers to absorption spectroscopy or reflect spectroscopy in the UV-VIS spectral region.
Ultraviolet-visible spectroscopy is an analytical method that can measure the amount of light received by the analyte.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
The use of Nauplii and metanauplii artemia in aquaculture (brine shrimp).pptxMAGOTI ERNEST
Although Artemia has been known to man for centuries, its use as a food for the culture of larval organisms apparently began only in the 1930s, when several investigators found that it made an excellent food for newly hatched fish larvae (Litvinenko et al., 2023). As aquaculture developed in the 1960s and ‘70s, the use of Artemia also became more widespread, due both to its convenience and to its nutritional value for larval organisms (Arenas-Pardo et al., 2024). The fact that Artemia dormant cysts can be stored for long periods in cans, and then used as an off-the-shelf food requiring only 24 h of incubation makes them the most convenient, least labor-intensive, live food available for aquaculture (Sorgeloos & Roubach, 2021). The nutritional value of Artemia, especially for marine organisms, is not constant, but varies both geographically and temporally. During the last decade, however, both the causes of Artemia nutritional variability and methods to improve poorquality Artemia have been identified (Loufi et al., 2024).
Brine shrimp (Artemia spp.) are used in marine aquaculture worldwide. Annually, more than 2,000 metric tons of dry cysts are used for cultivation of fish, crustacean, and shellfish larva. Brine shrimp are important to aquaculture because newly hatched brine shrimp nauplii (larvae) provide a food source for many fish fry (Mozanzadeh et al., 2021). Culture and harvesting of brine shrimp eggs represents another aspect of the aquaculture industry. Nauplii and metanauplii of Artemia, commonly known as brine shrimp, play a crucial role in aquaculture due to their nutritional value and suitability as live feed for many aquatic species, particularly in larval stages (Sorgeloos & Roubach, 2021).
3. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
3
Given the Linear Time-Invariant System
( ) 00 xxuBxAx =+=
and the Quadratic Cost Function:
[ ] TT
T
TT
c PPRRdt
u
x
RS
SP
uxJ =>=
= ∫
∞
&0
2
1
0
The Optimal Regulator that Minimizes the Cost Function Jc is given by the following
procedure:
Define the Hamiltonian of the Optimization Problem:
( ) [ ] ( )uBxA
u
x
RS
SP
uxuxH T
T
TT
++
= λλ
2
1
,,
The Euler-Lagrange Equations are:
( )λλ
λ
λ
λ
TTTT
T
T
T
BxSRuBxSuR
u
H
uSxPA
x
H
td
d
uBxA
H
td
xd
+−=⇒++=
∂
∂
=
−−−=
∂
∂
−=
+=
∂
∂
=
−1
0
4. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
4
( )λλ
λ
λ
λ
TTTT
T
T
T
BxSRuBxSuR
u
H
uSxPA
x
H
td
d
uBxA
H
td
xd
+−=⇒++=
∂
∂
=
−−−=
∂
∂
−=
+=
∂
∂
=
−1
0
or: ( )
( ) ( )
=
−−−−
−−
=
−−
−−
λλλ
x
A
x
SRBASRSP
BRBSRBAx
HTTT
TT
11
11
( )
( ) ( )
−−−−
−−
=
−−
−−
TTT
TT
H
SRBASRSP
BRBSRBA
A
11
11
:
We want to find X (t) such that: , then( ) ( ) ( )txtXt =λ
( ) ( )XBSRKxKxXBSRu TT
cc
TT
+=⇒−=+−= −− 11
:
5. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
5
( )
( ) ( )
=
−−−−
−−
=
−−
−−
λλλ
x
A
x
SRBASRSP
BRBSRBAx
HTTT
TT
11
11
We want to find X (t) such that: , then( ) ( ) ( )txtXt =λ
( ) ( ) ( ) ( ) ( )txtXtxtXt +=λ
( ) ( ) ( )[ ]xXBRBxSRBAXxXxXSRBAxSRSP TTTTT 1111 −−−−
−−+=−−−−
( ) ( ) ( )[ ] ( )txxSRSPXBRBXSRBAXXSRBAX TTTTT
∀=−+−−+−+ −−−−
01111
( ) ( ) ( )TTTTT
SRSPXBRBXSRBAXXSRBAX 1111 −−−−
−−+−−−−=
We obtain the following Differential Riccati Equation for :X
( ) ( ) ( )
[ ]
( )
( ) ( )
[ ]
( ) ( ) 0
0
1
11
11
1111
=+++−+=
−=
−−−−
−−
−=
−−+−−−−=
−
−−
−−
−−−−
PXBSRXBSAXXA
X
I
AIX
X
I
SRBASRSP
BRBSRBA
IX
SRSPXBRBXSRBAXXSRBA
TTTTTT
HTTT
TT
TTTTT
If a Steady-state Solution exists for t→∞ then and:0→X
Continuous
Algebraic
Riccati
Equation
(CARE)
6. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
Theorem 1 A Unique Stabilizing
( A+BKc=[A-BR-1
(ST
+BT
X)] is Stable)
Solution of CARE, X=XT
, is obtained iff:
a.(A,B) is Stabilizable
b.Re[λi(AH)]≠0 for all i=1,2,…,2n
The Unique Stabilizing Solution is denoted X=Ric [AH]
Proof
a. It is clear that the condition that (A,B) is Stabilizable is a necessary condition to
stabilize A+BKc.
b. Rewrite
( )
( ) ( )
( )
( )[ ]
−
+−−
−+−
=
−−−−
−−
=
−
−−
−−
−−
IX
I
XBSRBA
BRBXBSRBA
IX
I
SRBASRSP
BRBSRBA
A TTT
TTT
TTT
TT
H
0
0
0
:
1
11
11
11
If λi is an eigenvalue of AH, - λi is also; hence AH has n eigenvalues λi s.t. Re[λi] ≤ 0
and n eigenvalues λj s.t. Re[λj] ≥ 0. If AH has no eigenvalue of jω axis, then we can
find a solution X s.t. all the eigenvalues of A-BR-1
(ST
+BT
X)=A+BKc are the stable
eigenvalues of AH.
( ) ( ) 01
1
1 =+++−+ −
PXBSRXBSAXXA TTTTTT
7. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
Theorem 1 A Unique Stabilizing
( A+BKc=[A-BR-1
(ST
+BT
X)] is Stable)
Solution of CARE is obtained iff:
a.(A,B) is Stabilizable
b.Re[λi(AH)]≠0 for all i=1,2,…,2n
The Unique Stabilizing Solution is denoted X=Ric [AH]
Proof (continue -1)
c. Assume that there are two Stabilizing Solutions X1=X1
T
and X2 =X2
T
that satisfy the
CARE ( ) ( )
( ) ( ) 0
0
2
1
222
1
1
111
=+++−+
=+++−+
−
−
PXBSRXBSAXXA
PXBSRXBSAXXA
TTTTTT
TTTTTT
Subtracting those two equations we obtain:
( ) ( ) ( ) ( )
0
0
1
1
21
1
2
2
1
21
1
1
1
2121
1
2121
=−+
−−−−−−−+−
−−
−−−−
XBRBXXBRBX
XBRBXXBRBXSRBXXXXBRSAXXXXA
TT
TTTTT
or: ( )[ ] ( ) ( ) ( )[ ] 01
1
21212
1
=+−−+−+− −−
XBSRBAXXXXXBSRBA TTTTT
8. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
Theorem 1 A Unique Stabilizing
( A+BKc=[A-BR-1
(ST
+BT
X)] is Stable)
Solution of CARE is obtained iff:
a.(A,B) is Stabilizable
b.Re[λi(AH)]≠0 for all i=1,2,…,2n
The Unique Stabilizing Solution is denoted X=Ric [AH]
Proof (continue -2)
c. Assume that there are two Stabilizing Solutions X1=X1
T
and X2 =X2
T
that satisfy the
CARE
This is a Sylvester Matrix Equation and since both X1 and X2 are Stabilizing
Solutions we must have:
( )[ ] ( ) ( ) ( )[ ] 01
1
21212
1
=+−−+−+− −−
XBSRBAXXXXXBSRBA TTTTT
( )[ ]{ }
( )[ ]{ }
( )[ ]{ } ( )[ ]{ } jiXBSRBAXBSRBA
njXBSRBA
niXBSRBA TT
j
TT
iTT
j
TT
i
,0ReRe
,,10Re
,,10Re
2
1
1
1
2
1
1
1
∀≠+−++−⇒
=∀<+−
=∀<+− −−
−
−
λλ
λ
λ
Therefore the solution of the Sylvester Matrix Equation is Unique, and by
substitution we can see that X1 = X2 is the Unique Solution.
q.e.d.
9. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
Proof (1)
(1) The Matrix YT
Z is Symmetric.
(2) If Y-1
exists, the X=ZY-1
is the Solution of CARE such that the Matrix
((A-BR-1
ST
)-BR-1
BT
X) has the Eigenvalues λ1,λ2,…,λn. X is Symmetric.
(3) If (A,B) Stabilizable and Re[λi(AH)]≠0 for all i=1,2,…,2n (Theorem 1) Y-1
exists.
Theorem 2 Let the columns of the Matrix be the Eigenvectors
of AH corresponding to the Stable Eigenvalues λ1,λ2,…,λn, and J the corresponding
Jordan Matrix. Then
( )nxnnxn
RZYR
Z
Y
∈∈
,2
Define TTT
SRSPQBRBWSRBAE 111
:,:,: −−−
−==−=
( )
( ) ( )
−−
−
=
−−−−
−−
=
−−
−−
TTTT
TT
H
EQ
WE
SRBASRSP
BRBSRBA
A
11
11
:
=−−
=−
⇒
=
−−
−
⇒
=
JZZEQY
JYZWYE
J
Z
Y
Z
Y
EQ
WE
J
Z
Y
Z
Y
A TTHWe have
( ) ZYJZWZZEYZYJWZEYJYZWYE TTTTTTTTTTT
=−⇒=−⇒=−
JZYQYYZEYJZYZEYQYYJZZEQYY TTTTTTTTTT
−−=⇒=−−⇒=−−
10. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
Proof (1) (continue – 1)
Since –YT
QT
Y-ZT
WZ is symmetric so is (YT
Z)J+JT
(YT
Z) or
( ) ( )ZYJJZYZWZQYYZWZZYJJZYQYY
JZYQYYZEY
ZYJZWZZEY TTTTTTTTTT
TTTT
TTTTT
+=−−⇒+=−−⇒
−−=
=−
( ) ( ) ( ) ( )[ ] ( ) ( )TTTTTTTTTTT
ZYJJZYZYJJZYZYJJZY +=+=+
Because J represents the stable Eigenvalues Re[λi(J)+λj(J)]≠0 for all I,j=1,…,n, the
Unique Solution of the Sylvester Matrix Equation is
( ) ( )[ ] ( ) ( )[ ] [ ]0=−+−
TTTTTTT
ZYZYJJZYZY Sylvester Matrix Equation
( ) ( ) [ ] ( ) ( ) SymmetricisZYZYZYZYZY TTTTTTT
=⇒=− 0
(1) The Matrix YT
Z is Symmetric.
(2) If Y-1
exists, the X=ZY-1
is the Solution of CARE such that the Matrix
((A-BR-1
ST
)-BR-1
BT
X) has the Eigenvalues λ1,λ2,…,λn. X is Symmetric.
(3) If (A,B) Stabilizable and Re[λi(AH)]≠0 for all i=1,2,…,2n (Theorem 1) Y-1
exists.
Theorem 2 Let the columns of the Matrix be the Eigenvectors
of AH corresponding to the Stable Eigenvalues λ1,λ2,…,λn, and J the corresponding
Jordan Matrix. Then
( )nxnnxn
RZYR
Z
Y
∈∈
,2
11. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
Proof (2)
=−−
=−
JZZEQY
JYZWYE
T
111 −−−
=−⇒=− YJYYZWEYJYZWYE
1111111 −−−−−−−
=−⇒=− YJZYZWYZEYZYJYYZWEYZ
111 −−−
=−−⇒=−− YJZYZEQYJZZEQY TT
( ) ( ) ( ) ( ) [ ]01111
=+−+ −−−−
QYZWYZEYZYZET
Define TTT
SRSPQBRBWSRBAE 111
:,:,: −−−
−==−=
Therefore X=ZY-1
is the Unique Stabilizing Solution of the CARE
( ) ( ) ( ) [ ]01111
=−+−−+− −−−− TTTTT
SRSPXBRBXSRBAXXSRBA q.e.d.
(1) The Matrix YT
Z is Symmetric.
(2) If Y-1
exists, the X=ZY-1
is the Solution of CARE such that the Matrix
((A-BR-1
ST
)-BR-1
BT
X) has the Eigenvalues λ1,λ2,…,λn. X is Symmetric.
(3) If (A,B) Stabilizable and Re[λi(AH)]≠0 for all i=1,2,…,2n (Theorem 1) Y-1
exists.
Theorem 2 Let the columns of the Matrix be the Eigenvectors
of AH corresponding to the Stable Eigenvalues λ1,λ2,…,λn, and J the corresponding
Jordan Matrix. Then
( )nxnnxn
RZYR
Z
Y
∈∈
,2
Assume Y-1
exists:
12. Linear Quadratic Regulator (LQR) Problem
SOLO
Inner-Outer and Spectral Factorizations
Proof (2) (continue – 1)
q.e.d.
From (1) YT
Z is Symmetric, so
( ) YZZYZY TTTT
==
TTTTTTTT
ZYXZXYZYZYYYZZY −−−
=⇒=⇒=⇒= 11
1−
= YZX
( ) TTTT
XYZZYX === −− 1
X is Symmetric
(1) The Matrix YT
Z is Symmetric.
(2) If Y-1
exists, the X=ZY-1
is the Solution of CARE such that the Matrix
((A-BR-1
ST
)-BR-1
BT
X) has the Eigenvalues λ1,λ2,…,λn. X is Symmetric.
(3) If (A,B) Stabilizable and Re[λi(AH)]≠0 for all i=1,2,…,2n (Theorem 1) Y-1
exists.
Theorem 2 Let the columns of the Matrix be the Eigenvectors
of AH corresponding to the Stable Eigenvalues λ1,λ2,…,λn, and J the corresponding
Jordan Matrix. Then
( )nxnnxn
RZYR
Z
Y
∈∈
,2
13. References
SOLO
13
S. Hermelin, “Robustness and Sensitivity Design of Linear Time-Invariant Systems”,
PhD Thesis, Stanford University, 1986
G.Stein, J.Doyle, B.Francis, “Advances in Multivariable Control”, ONR/Honeywell
Workshop, 1984
Inner-Outer and Spectral Factorizations
T. Kailath, A.H. Sayed, B. Hassibi, “Linear Estimation”. Prentice Hall, 2000
14. 14
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA
15. Sylvester Matrix Equation : Anxn Xnxm+Xnxm Bmxm=Cnxm
SOLO
15
Consider the Sylvester Matrix Equation
where Anxn, Bmxm, Cnxm are given matrices. Then there
exists a Unique Solution Xnxm if and only if
Matrices
James Joseph Sylvester
(1814 – 1887)
nxmmxmnxmnxmnxn CBXXA =+
( )[ ] ( )[ ] njiAA nxnjnxni ,,1,0ReRe =∀≠+ λλ
∫
∞
=
0
dteCeX tB
nxm
tA
nxm
mxmnxn
Note : If B=AT
: Lyapunov Equation
The Necessary and Sufficient Condition for the existence of a
Unique Solution is
nxn
T
nxnnxnnxnnxn CAXXA =+
In particular if λi(A)=- λj(A) = j ω a Unique Solution
does not exist.
Aleksandr Mikhailovich
Lyapunov
1857 - 1918
( ) ( ) mjniBA mxmjnxni ,,1,,,10 ==∀≠+ λλ
the Unique Solution is given by
If ( )[ ] ( )[ ] mjniBA mxmjnxni ,,1,,,10ReRe ==∀<+ λλ
16. Sylvester Matrix Equation : Anxn Xnxm+Xnxm Bmxm=Cnxm
SOLO
Matrices
Proof
Let rewrite
[ ]0
21
22221
11211
21
22221
11211
21
22221
11211
21
22221
11211
21
22221
11211
=
−
+
=
−+
nmnn
m
m
mmmm
m
m
nmnn
m
m
nmnn
m
m
nnnn
n
n
nxmmxmnxmnxmnxn
ccc
ccc
ccc
bbb
bbb
bbb
xxx
xxx
xxx
xxx
xxx
xxx
aaa
aaa
aaa
CBXXA
=
nmnn
m
m
nnnn
n
n
nxmnxn
xxx
xxx
xxx
aaa
aaa
aaa
XA
21
22221
11211
21
22221
11211
( ) ( )
( )
[ ]nxmnxnc
mnxn
nxn
nxn
Xvec
m
nm
m
n
n
nxn
nxn
nxn
nxmcnxnm XAvec
cA
cA
cA
c
x
x
c
x
x
c
x
x
A
A
A
XvecAI
nxmc
=
=
=⊗
2
1
1
2
2
12
1
1
11
00
00
00
( ) [ ]
( )
( )
nxmc Xvec
xmn
m
nm
m
n
n
mcnxmc
c
x
x
c
x
x
c
x
x
cccvecXvec
1
1
2
2
12
1
1
11
21 :
⋅
==
Define the
Vectorization
Operator vec:
18. Sylvester Matrix Equation : Anxn Xnxm+Xnxm Bmxm=Cnxm
SOLO
Matrices
Proof (continue – 2)
We have
[ ] [ ] [ ] [ ] [ ]0=−+=−+ nxmcmxmnxmcnxmnxncnxmmxmnxmnxmnxnc CvecBXvecXAvecCBXXAvec
( ) ( ) [ ]nxmcnxmcn
T
mxmnxnm CvecXvecIBAI =⊗+⊗or
Let use the Jordan decomposition for Anxn and Bmxm
11
&
−−
== TTT
BBB
T
mxmAAAnxn SJSBSJSA
( ) ( )
( ) ( ) ( ) ( )
n
TTT
m
TT
TTT
I
AABBBAAA
I
BB
nBBBAAAmn
T
mxmnxnm
SSSJSSJSSS
ISJSSJSIIBAI
1111
11
−−−−
−−
⊗+⊗=
⊗+⊗=⊗+⊗
( )
( )( ) ( ) ( )
( ) ( )( )
( ) ( )
( )( )
( )( )
( )
( )
111
111
1111
−−−
−−−
⊗
−−
⊗⊗⊗⊗
−−
⋅⊗⋅=⊗⊗
⊗=⊗
⊗⊗+⊗⊗=
ATB
T
nTBATB
TT
ATBAm
TT
SS
AB
IJSS
ABB
SSJI
AABAB
DBCADCBA
BABA
SSSJSSJSSS
( )( )( ) 1−
⊗⊗+⊗⊗=⊗+⊗ ABnBAmABn
T
mxmnxnm SSIJJISSIBAI T
T
T
This equation has a Unique Solution iff is Nonsingular.( )n
T
mxmnxnm IBAI ⊗+⊗
19. Sylvester Matrix Equation : Anxn Xnxm+Xnxm Bmxm=Cnxm
SOLO
Matrices
Proof (continue – 3)
where
( )( )( ) 1−
⊗⊗+⊗⊗=⊗+⊗ ABnBAmABn
T
mxmnxnm SSIJJISSIBAI T
T
T
[ ] [ ] [ ]
[ ] [ ] [ ]
[ ] [ ] [ ]
=
lJ
J
J
J
00
00
00
2
1
[ ]
=
i
i
i
i
i
xkki ii
J
λ
λ
λ
λ
λ
0000
1000
0000
0010
0001
Ji are Upper-Triangular Matrices
( )nBAm IJJI T ⊗+⊗Therefore is also Upper Triangular with diagonal elements λi[A] + λj[B]
(i=1,…,n, j=1,…,m), that are the Eigenvalues of and therefore to the
Similar Matrix . This Matrix is Nonsingular iff
( )nBAm IJJI T ⊗+⊗
( )n
T
mxmnxnm IBAI ⊗+⊗
( ) ( ) mjniBA mxmjnxni ,,1,,,10 ==∀≠+ λλ
20. Sylvester Matrix Equation : Anxn Xnxm+Xnxm Bmxm=Cnxm
SOLO
Matrices
Since ( )[ ] ( )[ ] [ ] [ ]0lim,0lim,,1,,,10ReRe ====∀<+
∞→∞→
tB
t
tA
t
mxmjnxni eemjniBA λλ
Proof (continue – 4)
Let rewrite
[ ] [ ]
=
−
=−+=
nxm
m
Snnxm
nxm
m
nxnnxm
mxm
nnxmnxmmxmnxmnxmnxn
X
I
AIX
X
I
AC
B
IXCBXXA
0
0
( ) ( ) nxmnxm
tB
nxm
tA
nxm CPeCetP mxmnxn
−=⇒−= 0:Define
By differentiation
( )
mxmnxmnxmnxnmxm
tB
nxm
tAtB
nxm
tA
nxn
nxm
BPPABeCeeCeA
td
tPd mxmnxnmxmnxn
−−=−−=
Let Integrate the Differential Equation
( ) ( ) ( )
mxmnxmnxmnxn
nxm
nxmnxm BdtPdtPAdt
td
tPd
PP
−+
−==−∞ ∫∫∫
∞∞∞
000
0
We have
and
where
∫
∞
=
0
dteCeX tB
nxm
tA
nxm
mxmnxn
( ) [ ]0=∞nxmP
mxmnxmnxmnxnmxmnxmnxmnxnnxm BXXABdtPdtPAC +=
−+
−= ∫∫
∞∞
00
21. Matrix Differential Riccati Equation
SOLO
Inner-Outer and Spectral Factorizations
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tDtXtCtXtBtXtXtAtX +++=
where all matrices are nxn and A (t),B (t), C(t), D(t) are integrable over an
interval t0 ≤ t ≤ tf.
This Matrix Differential Equation can be decompose in the following 2 Linear
Differential Equations:
The Nonlinear Matrix Differential Riccati Equation is given by
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )tZtAtYtDtZ
tZtCtYtBtY
+=
−−=
If in the interval t0 ≤ t ≤ tf the Matrix Y(t) is nonsingular, then the solution of the
Riccati Differential Equation is ( ) ( ) ( ) 10
1
ttttYtZtX ≤≤= −
To verify we have ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tY
td
tYd
tYtY
td
d
tY
td
d
tZtY
td
tZd
tX
td
d 11111
& −−−−−
−=+=
( ) ( ) ( ) ( ) ( )[ ] ( ) ( ) ( ) ( ) ( ) ( ) ( )[ ] ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tXtCtXtBtXtXtAtD
tYtZtCtYtBtYtZtYtZtAtYtDtX
td
d
+++=
−−−+= −−− 111
22. Matrix Differential Riccati Equation
SOLO
Inner-Outer and Spectral Factorizations
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )tDtXtCtXtBtXtXtAtX +++=
If yhe matrices are nxn and A ,B , C, D are constant, then
This Matrix Differential Equation can be decompose in the following 2 Linear
Differential Equations:
The Nonlinear Matrix Differential Riccati Equation is given by
( ) ( ) ( )
( ) ( ) ( )tZAtYDtZ
tZCtYBtY
+=
−−=
If in the interval t0 ≤ t ≤ tf the Matrix Y(t) is nonsingular, then the solution of the
Riccati Differential Equation is ( ) ( ) ( ) 10
1
ttttYtZtX ≤≤= −
Define ( )
( )
( )
−−
=
=
AD
CB
G
tZ
tY
tW :&:
to obtain which have the solution( ) ( )tGWtW = ( ) ( )
( )0
0
tWetW ttG −
=
Editor's Notes
K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967 , pp.356-357
K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967 , pp.356-357
K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967 , pp.356-357
K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967 , pp.356-357
K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967 , pp.356-357
K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967 , pp.321-322
K. Ogata, “State Space Analysis of Control Systems”, Prentice Hall, Inc., 1967 , pp.321-322