time domain analysis, Rise Time, Delay time, Damping Ratio, Overshoot, Settli...Waqas Afzal
Time Response- Transient, Steady State
Standard Test Signals- U(t), S(t), R(t)
Analysis of First order system - for Step input
Analysis of second order system -for Step input
Time Response Specifications- Rise Time, Delay time, Damping Ratio, Overshoot, Settling Time
Calculations
time domain analysis, Rise Time, Delay time, Damping Ratio, Overshoot, Settli...Waqas Afzal
Time Response- Transient, Steady State
Standard Test Signals- U(t), S(t), R(t)
Analysis of First order system - for Step input
Analysis of second order system -for Step input
Time Response Specifications- Rise Time, Delay time, Damping Ratio, Overshoot, Settling Time
Calculations
CONTROL SYSTEMS PPT ON A LEAD COMPENSATOR CHARACTERISTICS USING BODE DIAGRAM ...sanjay kumar pediredla
A LEAD COMPENSATOR CHARACTERISTICS USING BODE DIAGRAM FOR MAXIMUM OF 50 DEG PHASE ANGLE
THIS PPT IS SO USEFUL FOR THE ENGINEERING STUDENTS FOR CONTROL SYSTEMS STUDENTS AND THIS PPT ALSO CONTAINS A MATLAB CODING FOR THE LEAD COMPENSATOR AND THE RESULTS ARE ALSO PLOTTED IN THAT PPT AND THE PROBLEM CAN ALSO BE SOLVED BY USING THE DATA IN PPT AND IT IS SO USEFUL PPT
Conversion of transfer function to canonical state variable modelsJyoti Singh
Realization of transfer function into state variable models is needed even if the control system design based on frequency-domain design method.
In these cases the need arises for the purpose of transient response simulation.
But there is not much software for the numerical inversion of Laplace transform.
So one ways is to convert transfer function of the system to state variable description and numerically integrating the resulting differential equations rather than attempting to compute the inverse Laplace transform by numerical method.
State space analysis, eign values and eign vectorsShilpa Shukla
State space analysis concept, state space model to transfer function model in first and second companion forms jordan canonical forms, Concept of eign values eign vector and its physical meaning,characteristic equation derivation is presented from the control system subject area.
Cylindrical and spherical coordinates shalinishalini singh
In this Presentation, I have explained the co-ordinate system in three plain. ie Cylindrical, Spherical, Cartesian(Rectangular) along with its Differential formulas for length, area &volume.
CONTROL SYSTEMS PPT ON A LEAD COMPENSATOR CHARACTERISTICS USING BODE DIAGRAM ...sanjay kumar pediredla
A LEAD COMPENSATOR CHARACTERISTICS USING BODE DIAGRAM FOR MAXIMUM OF 50 DEG PHASE ANGLE
THIS PPT IS SO USEFUL FOR THE ENGINEERING STUDENTS FOR CONTROL SYSTEMS STUDENTS AND THIS PPT ALSO CONTAINS A MATLAB CODING FOR THE LEAD COMPENSATOR AND THE RESULTS ARE ALSO PLOTTED IN THAT PPT AND THE PROBLEM CAN ALSO BE SOLVED BY USING THE DATA IN PPT AND IT IS SO USEFUL PPT
Conversion of transfer function to canonical state variable modelsJyoti Singh
Realization of transfer function into state variable models is needed even if the control system design based on frequency-domain design method.
In these cases the need arises for the purpose of transient response simulation.
But there is not much software for the numerical inversion of Laplace transform.
So one ways is to convert transfer function of the system to state variable description and numerically integrating the resulting differential equations rather than attempting to compute the inverse Laplace transform by numerical method.
State space analysis, eign values and eign vectorsShilpa Shukla
State space analysis concept, state space model to transfer function model in first and second companion forms jordan canonical forms, Concept of eign values eign vector and its physical meaning,characteristic equation derivation is presented from the control system subject area.
Cylindrical and spherical coordinates shalinishalini singh
In this Presentation, I have explained the co-ordinate system in three plain. ie Cylindrical, Spherical, Cartesian(Rectangular) along with its Differential formulas for length, area &volume.
Aerodynamics Part III of 3 describes aerodynamics of wings in supersonic 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.
Describes the Color Theory History.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
For more presentations on other subjects please 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.
Aerodynamics Part II of 3 describes aerodynamics of bodies in supersonic 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 term Diffraction has been defined by Sommerfield as any deviation of light rays from rectilinear paths which cannot be interpreted as reflection or refraction.
For comments please connect me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com. This presentation is in the Optics folder.
Presents the Tracking methods of moving targets by sensors (radar, electro optics,..).
For comments please contact me at solo.hermelin@gmail.com.
For more presentations visit my website at http://www.solohermelin.com.
I recommend to view this presentation on my website at RADAR folder, Tracking Systems subfolder.
Describes the general solutions of Electromagnetic Maxwell Equations.
Intended or Graduate Students in Science (math, physics, engineering) with previous knowledge in electromagnetics.
Please send me comments and suggestions for improvements to solo.hermelin@gmail.com.
More presentations can be found in my website at http://www.solohermelin.com.
The Controller Design For Linear System: A State Space ApproachYang Hong
The controllers have been widely used in many industrial processes. The goal of accomplishing a practical control system design is to meet the functional requirements and achieve a satisfactory system performance. We will introduce the design method of the state feedback controller, the state observer and the servo controller with optimal control law for a linear system in this paper.
The Design of Reduced Order Controllers for the Stabilization of Large Scale ...sipij
This paper investigates the design of reduced order controllers for the stabilization of large scale linear discrete-time control systems. Sufficient conditions are derived for the design of reduced order controllers by obtaining a reduced order model of the original large scale linear system using the dominant state of the system. The reduced order controllers are assumed to use only the state of the reduced order model of the original plant.
Describes Radar Tracking Loops in Range, Doppler and Angles.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://solohermelin.com.
Lyapunov-type inequalities for a fractional q, -difference equation involvin...IJMREMJournal
In this paper, we present new Lyapunov-type inequalities for a fractional boundary value problem of
fractional
q, -difference equation with p-Laplacian operator. The obtained inequalities are used to obtain a
lower bound for the eigenvalues of corresponding equations.
Digital Signal Processing[ECEG-3171]-Ch1_L03Rediet Moges
This Digital Signal Processing Lecture material is the property of the author (Rediet M.) . It is not for publication,nor is it to be sold or reproduced.
#Africa#Ethiopia
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.
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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.
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.
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
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.
Introduction:
RNA interference (RNAi) or Post-Transcriptional Gene Silencing (PTGS) is an important biological process for modulating eukaryotic gene expression.
It is highly conserved process of posttranscriptional gene silencing by which double stranded RNA (dsRNA) causes sequence-specific degradation of mRNA sequences.
dsRNA-induced gene silencing (RNAi) is reported in a wide range of eukaryotes ranging from worms, insects, mammals and plants.
This process mediates resistance to both endogenous parasitic and exogenous pathogenic nucleic acids, and regulates the expression of protein-coding genes.
What are small ncRNAs?
micro RNA (miRNA)
short interfering RNA (siRNA)
Properties of small non-coding RNA:
Involved in silencing mRNA transcripts.
Called “small” because they are usually only about 21-24 nucleotides long.
Synthesized by first cutting up longer precursor sequences (like the 61nt one that Lee discovered).
Silence an mRNA by base pairing with some sequence on the mRNA.
Discovery of siRNA?
The first small RNA:
In 1993 Rosalind Lee (Victor Ambros lab) was studying a non- coding gene in C. elegans, lin-4, that was involved in silencing of another gene, lin-14, at the appropriate time in the
development of the worm C. elegans.
Two small transcripts of lin-4 (22nt and 61nt) were found to be complementary to a sequence in the 3' UTR of lin-14.
Because lin-4 encoded no protein, she deduced that it must be these transcripts that are causing the silencing by RNA-RNA interactions.
Types of RNAi ( non coding RNA)
MiRNA
Length (23-25 nt)
Trans acting
Binds with target MRNA in mismatch
Translation inhibition
Si RNA
Length 21 nt.
Cis acting
Bind with target Mrna in perfect complementary sequence
Piwi-RNA
Length ; 25 to 36 nt.
Expressed in Germ Cells
Regulates trnasposomes activity
MECHANISM OF RNAI:
First the double-stranded RNA teams up with a protein complex named Dicer, which cuts the long RNA into short pieces.
Then another protein complex called RISC (RNA-induced silencing complex) discards one of the two RNA strands.
The RISC-docked, single-stranded RNA then pairs with the homologous mRNA and destroys it.
THE RISC COMPLEX:
RISC is large(>500kD) RNA multi- protein Binding complex which triggers MRNA degradation in response to MRNA
Unwinding of double stranded Si RNA by ATP independent Helicase
Active component of RISC is Ago proteins( ENDONUCLEASE) which cleave target MRNA.
DICER: endonuclease (RNase Family III)
Argonaute: Central Component of the RNA-Induced Silencing Complex (RISC)
One strand of the dsRNA produced by Dicer is retained in the RISC complex in association with Argonaute
ARGONAUTE PROTEIN :
1.PAZ(PIWI/Argonaute/ Zwille)- Recognition of target MRNA
2.PIWI (p-element induced wimpy Testis)- breaks Phosphodiester bond of mRNA.)RNAse H activity.
MiRNA:
The Double-stranded RNAs are naturally produced in eukaryotic cells during development, and they have a key role in regulating gene expression .
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Richard's aventures in two entangled wonderlandsRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
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.
3. SOLO
Reduced Order Observers for Linear Systems
∈∈∈+=
∈∈∈∈+=
pxmpxnpx
nxmnxnmxnx
RDRCRyuDxCy
RBRARuRxuBxAx
1
11
Plant:
We want to construct a Observer such that it’s output will asymptotically converge
to .x
xˆ
4. SOLO
Reduced Order Observers for Linear Systems
∈∈∈+=
∈∈∈∈+=
pxmpxnpx
nxmnxnmxnx
RDRCRyuDxCy
RBRARuRxuBxAx
1
11
Plant:
Assume: npCrank pxn ≤=
Find:
( )xnpn
RC −
⊥ ∈ such that:
nxn
C
C
⊥
is nonsingular.
Solution: Find the Singular Value Decomposition (SVD) of C
( )[ ] H
CpnpxCCpxn nxnpxppxp
VUC −Σ= 0
where H means Transpose of a matrix and complex conjugate of it’s elements, and:
nC
H
C
H
CCpC
H
C
H
CC IVVVVIUUUU ==== ;
( )
( ) ( )nxnnpxpp
ppC
diagIdiagI
diagpxp
1,,1,1,1,,1,1
0,,,, 2121
==
>≥≥≥=Σ σσσσσσ
( ) ( ) ( ) ( ) ( ) ( )
[ ] H
CCxppnCxnpn nxnpnxpnpnxpn
VUC −−⊥−−⊥
Σ= −−⊥ 0Then:
UC is any orthogonal matrix and ΣC is any non-zero diagonal matrix.
5. SOLO
Reduced Order Observers for Linear Systems
Define:
We have : x
C
C
p
uDy
=
−
⊥
( ) 1
: xpn
RpxCp −
⊥ ∈=
( )
−
=
−
= ⊥
−
⊥ p
uDy
CC
p
uDy
C
C
x ††
1
or : ( ) pCuDyCx
††
⊥+−=
where:
( ) nxpTT
RCCCC ∈=
−1† is the Right Pseudo-Inverse of C or pICC =†
( ) ( )pnnxTT
RCCCC −−
⊥⊥⊥⊥ ∈=
1† is the Right Pseudo-Inverse of C or pnICC −⊥⊥ =†
Then:
( ) ( )
( ) ( )
=
=
−−
−
⊥⊥⊥
⊥
⊥
⊥ pnxppn
pnpxp
I
I
CCCC
CCCC
CC
C
C
0
0
††
††
††
( ) nICCCC
C
C
CC =+=
⊥⊥
⊥
⊥
††††
6. SOLO
Reduced Order Observers for Linear Systems
We have:
+=
+=
uDxCy
uBxAx
( ) pCuDyCx
††
⊥+−=and:xCp ⊥=
( ) ( )[ ]{ }uBpCuDyCACuBxACxCp ++−=+== ⊥⊥⊥⊥
††
or:
( ) uBCuDyCACpCACp ⊥⊥⊥⊥ +−+= ††
We want to obtain an estimation of . If we add we can see that:ppˆ ( )uBxCyL −−
( )[ ]
( )
( )
0ˆ
ˆˆ
0
††
††
=−−−−=
−+−−=−−
−
⊥
⊥
uDpCCuDyCCy
uDpCuDyCCyuDxCy
pnpxpI
Apparently does not contain any information on , but let compute .py y
( ) ( )[ ]{ } uDuBpCuDyCACuDuBxACuDxCy +++−=++=+= ⊥
ˆ††
7. SOLO
Reduced Order Observers for Linear Systems
We have:
Therefore contains the information on .py
( ) ( )[ ]{ } uDuBpCuDyCACuDuBxACuDxCy +++−=++=+= ⊥
ˆ††
( ) uDuBCpCACuDyCACy +++−= ⊥
ˆ††
Let estimate by using:p
( )
( )[ ]uDuBCpCACuDyCACyL
uBCuDyCACpCACp
−−−−−+
+−+=
⊥
⊥⊥⊥⊥
ˆ
ˆˆ
††
††
or:
( )[ ] ( ) ( )[ ]uBuDyCApCACLCuDyLp
td
d
+−+−=−− ⊥⊥
†† ˆˆ
( ) pCuDyCx
††
⊥+−=
8. SOLO
Reduced Order Observers for Linear Systems
We have:
( )[ ] ( ) ( )[ ]uBuDyCApCACLCuDyLp
td
d
+−+−=−− ⊥⊥
†† ˆˆ
( ) pCuDyCx ˆˆ ††
⊥+−=
9. SOLO
Reduced Order Observers for Linear Systems
We also have:
( )[ ] ( ) [ ]uBxACLCuDyLp
td
d
+−=−− ⊥
ˆˆ
( ) pCuDyCx ˆˆ ††
⊥+−=
10. SOLO
Reduced Order Observers for Linear Systems
One other
form: ( )[ ] ( )
( ) ( ) ( ) uBCLCuDyCACLC
pCACLCuDyLp
td
d
−+−−+
−=−−
⊥⊥
⊥⊥
†
† ˆˆ
( ) pCuDyCx ˆˆ ††
⊥+−=
11. SOLO
Reduced Order Observers for Linear Systems
And
another
form:
( )[ ] ( ) ( )[ ]
( ) ( )( ) ( ) uBCLCuDyLCCACLC
uDyLpCACLCuDyLp
td
d
−+−+−+
−−−=−−
⊥⊥⊥
⊥⊥
††
† ˆˆ
( )[ ] ( )( )uDyLCCuDyLpCx −++−−= ⊥⊥
††† ˆˆ
12. SOLO
Reduced Order Observers for Linear Systems
We have:
( )
( )[ ]uDuBCpCACuDyCACyL
uBCuDyCACpCACp
−−−−−+
+−+=
⊥
⊥⊥⊥⊥
ˆ
ˆˆ
††
††
Subtract those equations:
Define the estimation error:
( )
( )[ ]uDuBCpCACuDyCACyL
uBCuDyCACpCACp
−−−−−+
+−+=
⊥
⊥⊥⊥⊥
††
††
( ) ( )ppCACLppCACpp ˆˆˆ ††
−−−=− ⊥⊥⊥
ppp ˆ:~ −=
( ) pCACLCp ~~ †
⊥⊥ −=
p~We can see that ( the estimation error) is uncontrollable and is stable iff.
( )[ ] iCACLCi ∀<− ⊥⊥ 0Real
†
λ ppp →→ ˆ&0~
13. SOLO
Reduced Order Observers for Linear Systems
Note:
Define:
( )
( )
( ) ( ) ( )
( )
( ) ( ) ( )
H
C
pnxpnCxppn
pnpxC
pnxpnCxppn
pnpxC
xnpn
pxn
nxn
pxppxp
V
U
U
C
C
Σ
Σ
=
−−−
−
−−−
−
−⊥ ⊥⊥
0
0
0
0
( )
( )
( ) ( ) ( ) ( ) ( ) ( )
−
=
−
=
−−⊥−⊥−−−
−
− pxnxppnxnpn
pxn
xnpn
pxn
pnxpnxppn
pnpx
xnpn
pxn
CLC
C
C
C
IL
I
T
C pxp
0
:
Since:
( )
[ ]
( ) ( )
=
− −−
⊥
⊥− pn
p
pn
p
pn
p
I
I
IL
I
CC
C
C
IL
I
0
000 ††
Define:
[ ] [ ]
( )
[ ]†††††
0
: ⊥⊥
−
⊥ +=
= CLCC
IL
I
CCMH
pn
p
14. SOLO
Reduced Order Observers for Linear Systems
Note (continue – 1):
Define: CLCT −= ⊥:
Then:
( )[ ] ( ) ( )[ ]
( ) ( )( ) ( ) uBCLCuDyLCCACLC
uDyLpCACLCuDyLp
td
d
−+−+−+
−−−=−−
⊥⊥⊥
⊥⊥
††
† ˆˆ
( )[ ] ( )( )uDyLCCuDyLpCx −++−−= ⊥⊥
††† ˆˆ
[ ] [ ]†††
: ⊥⊥+= CLCCMH
( )uDyLpz −−= ˆ:
( )
( )
−+=
+−+=
uDyHzMx
uBTuDyHATzMAT
td
zd
KF
ˆ
( )
( )
( )
( ) AT
C
T
HMAT
C
T
HATMAT
I
T
C
MH
I
I
MH
T
C
KF
n
pn
p
=
=
=
=
−
0
0
Those are the well known
Reduced Order Observer
Equations
15. SOLO
Reduced Order Observers for Linear Systems
Note (continue – 2):
Then:
( )
( )
−+=
+−+=
uDyHzMx
uBTuDyHATzMAT
td
zd
GF
ˆ
( )
( )
( )
( ) AT
C
T
HMAT
C
T
HATMAT
I
T
C
MH
I
I
MH
T
C
GF
n
pn
p
=
=
=
=
−
0
0
( ) CGCHATTMIAT
TMATATTFAT
DHSuSyHzMx
DGBTJuJyGzF
td
zd
n ==−=
−=−
−=++=
−=++=
:ˆ
:
=+
=+
−=
=−
−
0DHS
ITMCH
DGBTJ
CGTFAT
valueseigenstablehasF
n
nxpnxmnxq
qxpqxmqxq
nq
HSM
GJF
xz
xx
yHuSzMx
yGuJzFz
RRR
RRR
RR
∈∈∈
∈∈∈
∈∈
→
++=
++=
,,
,,
,ˆ,
ˆ
ˆ
16. SOLO
Observers
Generic Observer for a Linear Time Invariant (LTI) System
pxmpxnnxmnxn
pmn
DCBA
yux
uDxCy
uBxAx
RRRR
RRR
∈∈∈∈
∈∈∈
+=
+=
,,,
,,
Observer
nxpnxmnxq
qxpqxmqxq
nq
RSM
GJF
xz
xx
yRuSzMx
yGuJzFz
RRR
RRR
RR
∈∈∈
∈∈∈
∈∈
→
++=
++=
,,
,,
,ˆ,
ˆ
ˆ
A Necessary Condition for obtaining an Observer is that (A,C) is Observable.
The Observer will achieve
if and only if:
xx →ˆ
=+
=+
−=
=−
−
0DRS
ITGCR
DGBTJ
CGTFAT
valueseigenstablehasF
n
L.T.I. System
[ ] [ ]†††
: ⊥⊥+= CLCCMH CLCT −= ⊥:
HATGMATF == :&:
17. SOLO
Reduced Order Observers for Linear Systems
Let use a constant feedback from the
Reduced Order Observer to control the plant:
xK ˆ
The control is
xKrpCKxKru ˆ~†
−=+−= ⊥
( )[ ]
( ) ( )[ ]ppCpCuDyCKr
pCuDyCKrxKru
ˆ
ˆˆ
†††
††
−−+−−=
+−−=−=
⊥⊥
⊥
The augmented system is
( )
[ ]
[ ]
+
−=+=
−+=+−=
+
−
=
⊥
⊥⊥
⊥⊥
rD
p
x
CKKDCuDxCy
p
x
CKKrpCKxKru
u
B
p
x
CACLC
A
p
x
†
††
†
~
00
0
~
18. SOLO
Reduced Order Observers for Linear Systems
The augmented system is
( )
[ ]
[ ]
+
−=
+
−
+
−
=
⊥
⊥
⊥⊥
rD
p
x
CKKDCy
r
B
p
x
CKK
B
CACLC
A
p
x
†
†
†
000
0
~
or
The poles of the closed loop system are given by:
( )
[ ]
+
−=
+
−
−
=
⊥
⊥⊥
⊥
rD
p
x
CKKDCy
r
B
p
x
CACLC
CKBKBA
p
x
†
†
†
00~
( ) ( )
[ ] ( ) ( )[ ]
ObservertheofPoles
pn
ControllertheofPoles
n
pn
n
CACLCIsKBAIs
CACLCIs
CKBKBAIs †
†
†
detdet
0
det ⊥⊥−
⊥⊥−
⊥
−−⋅+−=
−−
−+−
Hence the Reduced Order Controller has the “Separation Property” of the Controller and
Observer.
19. SOLO
Reduced Order Observers for Linear Systems
Compensator Transfer Function
By tacking the Laplace Transform of the compensator dynamics we obtain:
( )[ ] ( ) [ ]uBxACLCuDyLp
td
d
+−=−− ⊥
ˆˆ
( ) pCuDyCx ˆˆ ††
⊥+−=
xKu ˆ−=
( ) ppCCuDyCCxC
I
ˆˆˆ †
0
†
=+−= ⊥⊥⊥⊥
( )[ ] ( ) ( ) xKBACLCuDyLxCs ˆˆ −−=−− ⊥⊥
( ) ( ) ( )[ ]( ) xKBACLCKDLCsyLs xnpn
ˆ
−⊥⊥ −−−−=
( ) ( ) ( )[ ] ( ) ( ) 1
†ˆ mxxmpnpnnx yLsKBACLCKDLCsx −−⊥⊥ −−−−=
where
Therefore
( ) ( ) ( )[ ]( ) ( ) ( ) ( )[ ] ( ) ( )pnpnnxxnpn
IKBACLCKDLCsKBACLCKDLCs −−⊥⊥−⊥⊥ =−−−−−−−−
†
( ) ( ) ( )[ ] ( ) ( ) 1
†
mxxmpnpnnx yLsKBACLCKDLCsKu −−⊥⊥ −−−−−=
20. SOLO
Reduced Order Observers for Linear Systems
How to Find K & L For a Stable Closed Loop
This can be done by solving the following
Asymptotic Minimum Variance Control Problem:
xKu ˆ−=
( ) ( ){ } ( ) ( ) ( ){ }
+=
−===++=
uDxCy
tWwtwEtwExwuBxAx T
τδτ 0,0,00
( ) ( ) ( ) ( ){ } )(0lim definitepositiveRtuRtutxQtxEJ TT
t
>+=
∞→
System with no output noise to allow us to use a Reduced Order Observer.
The solution to this problem is:
where: PBRK T1−
=
and P is the solution of the Algebraic Riccati Equation:
01
=−++ −
PBRBPQAPPA TT or:
HT
T
ARic
AQ
BRBA
RicP =
−−
−
=
−1
Minimize:
A stabilizing solution (and unique) exists iff:
1 (A,B) is stabilizable
2 AH has no jω axis eigenvalues
If Q ≥ 0 then P ≥ 0
21. SOLO
Reduced Order Observers for Linear Systems
How to Find K & L For a Stable Closed Loop (continue – 1)
( ) wCuBCuDyCACpCACp
inputknown
⊥⊥⊥⊥⊥ ++−+=
1
††
( ) pCuDyCx
††
⊥+−=and:xCp ⊥=
and
( )wuBxACxCp ++== ⊥⊥
( ) ( )[ ]{ }
( ) wCuDuBCuDyCACpCAC
uDwuBpCuDyCACuDwuBxACuDxCy
inputknown
+++−+=
++++−=+++=+=
⊥
⊥
2
††
†† ˆ
The measurements are given by (instead of )y y
Let define:
†*†*
****
:,:
:,:,:,:
⊥⊥⊥
⊥
==
====
CACCCACA
wCvwCwyypx
22. SOLO
Reduced Order Observers for Linear Systems
How to Find K & L For a Stable Closed Loop (continue – 2)
( ) wCuBCuDyCACpCACp
inputknown
⊥⊥⊥⊥⊥ ++−+=
1
††
( ) wCuDuBCuDyCACpCACy
inputknown
+++−+= ⊥
2
††
Define:
†*†*
****
:,:
:,:,:,:
⊥⊥⊥
⊥
==
====
CACCCACA
wCvwCwyypx
The Estimation Problem becomes:
( ) ∗∗∗∗
++= winputknownxAx 1
( ) ∗∗∗∗
++= vinputknownxCy 2
[ ]
=
=
⊥
⊥⊥⊥
**
**
00
00**
*
*
:
RS
SP
CWCCWC
CWCCWC
vw
v
w
E TTT
TT
TT
23. SOLO
Reduced Order Observers for Linear Systems
How to Find K & L For a Stable Closed Loop (continue – 3)
The Estimation Problem:
( ) ∗∗∗∗
++= winputknownxAx 1
( ) ∗∗∗∗
++= vinputknownxCy 2
[ ]
=
=
⊥
⊥⊥⊥
**
**
00
00**
*
*
:
RS
SP
CWCCWC
CWCCWC
vw
v
w
E TTT
TT
TT
The Solution to the Estimation Problem is:
( )( ) ( )[ ]( ) 1
0
†
0
1 −
⊥⊥
−∗∗∗
+=+= TTTT
CWCCACYCWCRCYSL
or
( )[ ] ( ) 1
0
†
0
−
⊥⊥ += TTT
CWCCCAYWCL
where
( )[ ] ( )
( )( ) ( )[ ]
−−−−
−−
=
∗−∗∗∗∗−∗∗∗
∗−∗∗∗−∗∗∗
CRSASRSP
CRCCRSA
RicY
T
TT
11
11
24. SOLO
Reduced Order Observers for Linear Systems
How to Find K & L For a Stable Closed Loop (continue – 4)
In explicit form (the Algebraic Riccati Equation) is:
But
( )[ ] ( )[ ] ( ) ( )( ) 0
1111
=−+−−+− ∗−∗∗∗∗−∗∗∗−∗∗∗∗−∗∗∗ TTT
SRSPYCRCYCRSAYYCRSA
( ) ( )
( ) ( )[ ] ( )pnnx
TT
nxnpn
TT
CACCWCCWIC
CACCWCCWCCACCRSA
−⊥
−
−⊥
⊥
−
⊥⊥⊥
∗−∗∗∗
−=
−=−
†1
00
†1
00
†1
( ) ( ) ( )
( ) ( ) †1
0
†
†1
0
†1
⊥
−
⊥
⊥
−
⊥
∗−∗∗
=
=
CACCWCCAC
CACCWCCACCRC
TTTT
TT
( ) ( )( ) ( )
( )[ ] TTT
n
TTTTT
CWCCWCCWIC
CWCCWCCWCCWCSRSP
⊥
−
⊥
−
⊥⊥⊥
∗−∗∗∗
−=
−=−
0
1
00
0
1
000
1
25. SOLO
Reduced Order Observers for Linear Systems
How to Find K & L For a Stable Closed Loop (continue – 5)
Therefore Y(n-p)x(n-p) is given by the following (n-p) Algebraic Riccati Equation:
( )[ ] ( )[ ]{ }T
TT
n
TT
n CACCWCCWICYYCACCWCCWIC
†1
00
†1
00 ⊥
−
⊥⊥
−
⊥ −+−
( ) ( ) YCACCWCCACY TTTT †1
0
†
⊥
−
⊥− ( )[ ] 00
1
00 =−+ ⊥
−
⊥
TTT
n CWCCWCCWIC
Note:
1 ( )[ ] ( ) ( ) ( )
( ) PCCWCCWI
CCWCCCWCWCCWCCWCCWICCWCCWIP
TT
n
T
I
TTTTT
n
TT
n
=−=
+−=−=
−
−−−−
1
00
1
00
1
00
1
00
21
00
2
2:
This is a Projection, since P2
= P, but oblique because P is
not symmetrical.
2 For W0 = In we get: ( ) ( ) ⊥⊥
−−
=−=−=− CCCCICCCCICCWCCWI n
TT
n
TT
n
††11
00
( ) ( ) ( ) ††††1
0
†
⊥⊥⊥
−
⊥ = CACCACCACCWCCAC TTTTTT
( )[ ] ††††1
00 ⊥⊥⊥⊥⊥⊥⊥
−
⊥ ==−
−
CACCACCCCACCWCCWIC
pnI
TT
n
( )[ ] TT
I
TTT
n CCCCCCCWCCWCCWIC ⊥⊥⊥⊥⊥⊥⊥
−
⊥ ==−
†
0
1
00
26. SOLO
Reduced Order Observers for Linear Systems
How to Find K & L For a Stable Closed Loop (continue – 6)
Hence, for W0=In, Y(n-p)x(n-p) is given by the following (n-p) Algebraic Riccati Equation:
Notice (continue – 1):
3 With this L , A*- L C* will have stable eigenvalues, but
2
( ) ( ) ( ) 0
†††††
†
=+−+ ⊥⊥⊥
−
⊥⊥⊥⊥⊥
⊥⊥
T
CCI
TTT
CCYCACCACYCACYYCAC
n
( )[ ] ( ) ( ) ††
0
1†
†
†
CACYCCCCAYCL TTCC
C
TTT
⊥
=
−
⊥⊥
⊥⊥
=+=
( ) †††
** ⊥⊥⊥⊥⊥ −=−=− CACLCCACLCACCLA
Therefore has stable eigenvalues, and the
Reduced Order Estimator is stable
( ) †
⊥⊥ − CACLC
27. SOLO
Reduced Order Observers for Linear Systems
How to Find K & L For a Stable Closed Loop (continue – 7)
Notice (continue – 2):
4 Following P.J. Blanvillain and T.L. Johnson
(IEEE Tr. AC., Vol. AC-23, No.1, June 1978) this
Problem is equivalent to the following
( ) ( )
=
+=
xCy
WNxuBxAx 0,0~0
Given
Find the Dynamic Compensator Parameters (F, G, H, M)
+=
+=
yMzHu
yGzFz
Compensator
Which minimizes the Quadratic Performance Index:
( ) ( ) ( ) ( ) ( )[ ]
+= ∫
∞
0
,,, dttuRtutxQtxEMHGFJ TT
28. SOLO
Reduced Order Observers for Linear Systems
Let append to the Reduced Order Observer the Stable Transfer Matrix
( ) ( )
=+−=
−
DC
BA
DBAIsCsQ
ˆˆ
ˆˆ
:ˆˆˆˆ 1
=−−− uBCpCACyCACy ˆ††
The input to the Stable Transfer Function will be the same
as for the Reduced Order Observer.
29. References
SOLO
Kwakernaak, H., Sivan, R., “Linear Optimal Control Systems”, Wiley Inter-science,
1972, pg.335
Reduced Order Observers for Linear Systems
Gelb A. Ed, “Applied Optimal Estimation”, The Analytic Science Corporation, 1974,
pg.320
30. August 13, 2015 30
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