state space representation,State Space Model Controllability and Observability Derive Transfer Function from State Space Equation Time Response and State Transition Matrix Eigen Value
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Block Diagram Representation Of State Space Model
Controllability and Observability
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
Eigen Value
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
Ch2 mathematical modeling of control system Elaf A.Saeed
Chapter 2 Mathematical modeling of control system From the book (Ogata Modern Control Engineering 5th).
2-1 introduction.
2-2 transfer function and impulse response function.
2-3 automatic control systems.
It gives how states are representing in various canonical forms and how it it is different from transfer function approach. and finally test the system controllability and observability by kalman's test
Ch2 mathematical modeling of control system Elaf A.Saeed
Chapter 2 Mathematical modeling of control system From the book (Ogata Modern Control Engineering 5th).
2-1 introduction.
2-2 transfer function and impulse response function.
2-3 automatic control systems.
state space modeling of electrical systemMirza Baig
Introduction
As systems become more complex, representing them with differential equations or transfer functions becomes cumbersome. This is even more true if the system has multiple inputs and outputs. This document introduces the state space method which largely alleviates this problem. The state space representation of a system replaces an nth order differential equation with a single first order matrix differential equation. The state space representation of a system is given by two equations :
The first equation is called the state equation, the second equation is called the output equation. For an nth order system (i.e., it can be represented by an nth order differential equation) with r inputs and m outputs the size of each of the matrices is as follows:
Several features:The state equation has a single first order derivative of the state vector on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.The output equation has the output on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.
q is nx1 (n rows by 1 column)q is called the state vector, it is a function of timeA is nxn; A is the state matrix, a constantB is nxr; B is the input matrix, a constant u is rx1; u is the input, a function of time C is mxn; C is the output matrix, a constant D is mxr; D is the direct transition matrix, a constant y is mx1; y is the output, a function of time
Derivation of of State Space Model (Electrical)
To develop a state space system for an electrical system, they choosing the voltage across capacitors, and current through inductors as state variables. Recall that
so if we can write equations for the voltage across an inductor, it becomes a state equation when we divide by the inductance (i.e., if we have an equation for einductor and divide by L, it becomes an equation for diinductor/dt which is one of our state variable). Likewise if we can write an equation for the current through the capacitor and divide by the capacitance it becomes a state equation for ecapacitor
There are three energy storage elements, so we expect three state equations. Try choosing i1, i2 and e1 as state variables. Now we want equations for their derivatives. The voltage across the inductor L2 is e1 (which is one of our state variables)so our first state variable equation is
This equation has our input (ia) and two state variable (iL2 and iL1) and the current through the capacitor. So from this we can get our second state equation
Our third, and final, state equation we get by writing an equation for the voltage across L1 (which is e2) in terms of our other state variables
references:
http://lpsa.swarthmore.edu/Representations/SysRepSS.html
https://en.wikipedia.org/wiki/State-space_representation
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.
Transfer Function and Mathematical Modeling
Transfer Function
Poles And Zeros of a Transfer Function
Properties of Transfer Function
Advantages and Disadvantages of T.F.
Translation motion
Rotational motion
Translation-Rotation counterparts
Analogy system
Force-Voltage analogy
Force-Current Analogy
Advantages
Example
This presention is about one of the most important concept of control system called State space analysis. Here basic concept of control system and state space are discussed.
Ch5 transient and steady state response analyses(control)Elaf A.Saeed
Chapter 5 Transient and steady-state response analyses. From the book (Ogata Modern Control Engineering 5th).
5-1 introduction.
5-2 First-Order System.
5-3 second-order system.
5-6 Routh’s stability criterion.
5-8 Steady-state errors in unity-feedback control systems.
Mr. C.S.Satheesh, M.E.,
Frequency response analysis
Frequency Domain Specifications
Resonant Peak Mr
Resonant Frequency ωr
Bandwidth ωh
Cut – off Rate
Gain margin Kg
Phase margin γ
POLAR PLOT
Bode PLOT
state space modeling of electrical systemMirza Baig
Introduction
As systems become more complex, representing them with differential equations or transfer functions becomes cumbersome. This is even more true if the system has multiple inputs and outputs. This document introduces the state space method which largely alleviates this problem. The state space representation of a system replaces an nth order differential equation with a single first order matrix differential equation. The state space representation of a system is given by two equations :
The first equation is called the state equation, the second equation is called the output equation. For an nth order system (i.e., it can be represented by an nth order differential equation) with r inputs and m outputs the size of each of the matrices is as follows:
Several features:The state equation has a single first order derivative of the state vector on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.The output equation has the output on the left, and the state vector, q(t), and the input u(t) on the right. There are no derivatives on the right hand side.
q is nx1 (n rows by 1 column)q is called the state vector, it is a function of timeA is nxn; A is the state matrix, a constantB is nxr; B is the input matrix, a constant u is rx1; u is the input, a function of time C is mxn; C is the output matrix, a constant D is mxr; D is the direct transition matrix, a constant y is mx1; y is the output, a function of time
Derivation of of State Space Model (Electrical)
To develop a state space system for an electrical system, they choosing the voltage across capacitors, and current through inductors as state variables. Recall that
so if we can write equations for the voltage across an inductor, it becomes a state equation when we divide by the inductance (i.e., if we have an equation for einductor and divide by L, it becomes an equation for diinductor/dt which is one of our state variable). Likewise if we can write an equation for the current through the capacitor and divide by the capacitance it becomes a state equation for ecapacitor
There are three energy storage elements, so we expect three state equations. Try choosing i1, i2 and e1 as state variables. Now we want equations for their derivatives. The voltage across the inductor L2 is e1 (which is one of our state variables)so our first state variable equation is
This equation has our input (ia) and two state variable (iL2 and iL1) and the current through the capacitor. So from this we can get our second state equation
Our third, and final, state equation we get by writing an equation for the voltage across L1 (which is e2) in terms of our other state variables
references:
http://lpsa.swarthmore.edu/Representations/SysRepSS.html
https://en.wikipedia.org/wiki/State-space_representation
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.
Transfer Function and Mathematical Modeling
Transfer Function
Poles And Zeros of a Transfer Function
Properties of Transfer Function
Advantages and Disadvantages of T.F.
Translation motion
Rotational motion
Translation-Rotation counterparts
Analogy system
Force-Voltage analogy
Force-Current Analogy
Advantages
Example
This presention is about one of the most important concept of control system called State space analysis. Here basic concept of control system and state space are discussed.
Ch5 transient and steady state response analyses(control)Elaf A.Saeed
Chapter 5 Transient and steady-state response analyses. From the book (Ogata Modern Control Engineering 5th).
5-1 introduction.
5-2 First-Order System.
5-3 second-order system.
5-6 Routh’s stability criterion.
5-8 Steady-state errors in unity-feedback control systems.
Mr. C.S.Satheesh, M.E.,
Frequency response analysis
Frequency Domain Specifications
Resonant Peak Mr
Resonant Frequency ωr
Bandwidth ωh
Cut – off Rate
Gain margin Kg
Phase margin γ
POLAR PLOT
Bode PLOT
Modern Control - Lec 05 - Analysis and Design of Control Systems using Freque...
Similar to state space representation,State Space Model Controllability and Observability Derive Transfer Function from State Space Equation Time Response and State Transition Matrix Eigen Value
Sensors by computer applications gghhhhhhhhhhhhhhhhhhhhhhffffffffffffffffffffffffffffgffggttttsvbkihhhggttfffffffffffffffffggggttggggyyyyy h ilc you k fd7ihft I yft8iggd e u hbvd67ufr6uufffhjoytrdfhiigfffghj I cc c for judging Jukka c ffffg b gffryhhu.
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State-Space Analysis of Control System: Vector matrix representation of state equation, State transition matrix, Relationship between state equations and high-order differential equations, Relationship between state equations and transfer functions, Block diagram representation of state equations, Decomposition Transfer Function, Kalman’s Test for controllability and observability
Similar to state space representation,State Space Model Controllability and Observability Derive Transfer Function from State Space Equation Time Response and State Transition Matrix Eigen Value (20)
Why Fourier Transform
General Properties & Symmetry relations
Formula and steps
magnitude and phase spectra
Convergence Condition
mean-square convergence
Gibbs phenomenon
Direct Delta
Energy Density Spectrum
. Types of Modulation(Analog)
Phase-Frequency Relationships
FM and PM basics
Frequency deviation
MODULATION INDEX
Classification of FM
Narrow Band FM (NBFM)
generating a narrowband FM signal.
Wide Band FM (WBFM).
Carson’s Rule
Generation of WBFM
Average Power
FM BANDWIDTH
Comparing Frequency Modulation to Phase Modulation
ROOT-LOCUS METHOD, Determine the root loci on the real axis /the asymptotes o...Waqas Afzal
Angle and Magnitude Conditions
Example of Root Locus
Steps
constructing a root-locus plot is to locate the open-loop poles and zeros in s-plane.
Determine the root loci on the real axis
Determine the asymptotes of the root loci
Determine the breakaway point.
Closed loop stability via root locus
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
introduction to modeling, Types of Models, Classification of mathematical mod...Waqas Afzal
Types of Systems
Ways to study system
Model
Types of Models
Why Mathematical Model
Classification of mathematical models
Black box, white box, Gray box
Lumped systems
Dynamic Systems
Simulation
laplace transform and inverse laplace, properties, Inverse Laplace Calculatio...Waqas Afzal
Laplace Transform
-Proof of common function
-properties
-Initial Value and Final Value Problems
Inverse Laplace Calculations
-by identification
-Partial fraction
Solution of Ordinary differential using Laplace and inverse Laplace
Transfer Function, Concepts of stability(critical, Absolute & Relative) Poles...Waqas Afzal
Transfer Function
The Order of Control Systems
Concepts of stability(critical, Absolute & Relative)
Poles, Zeros
Stability calculation
BIBO stability
Transient Response Characteristics
Signal Flow Graph, SFG and Mason Gain Formula, Example solved with Masson Gai...Waqas Afzal
Basic Properties of SFG
Definitions of SFG Terms
SFG Algebra
Relation between SFG and block diagram
Mason Gain Formula
Example solved with Masson Gain Formula
automatic control, Basic Definitions, Classification of Control systems, Requ...Waqas Afzal
Why automatic controls is required
2. Process Variables
controlled variable, manipulated variable
3. Functions of Automatic Control
Measurement
Comparison
Computation
Correction
4.Basic Definitions
System, Plant, Process, Controller, input, output, disturbance
5. Classification of Control systems
Natural, Manmade & Automatic control system
Open-Loop, Close-Loop control System
Linear Vs Nonlinear System
Time invariant vs Time variant
Continuous Data Vs Discrete Data System
Deterministic vs Stochastic System
6. Requirements of an ideal Control system
Accuracy, Sensitivity, noise, Bandwidth, Speed, Oscillations
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Explore the innovative world of trenchless pipe repair with our comprehensive guide, "The Benefits and Techniques of Trenchless Pipe Repair." This document delves into the modern methods of repairing underground pipes without the need for extensive excavation, highlighting the numerous advantages and the latest techniques used in the industry.
Learn about the cost savings, reduced environmental impact, and minimal disruption associated with trenchless technology. Discover detailed explanations of popular techniques such as pipe bursting, cured-in-place pipe (CIPP) lining, and directional drilling. Understand how these methods can be applied to various types of infrastructure, from residential plumbing to large-scale municipal systems.
Ideal for homeowners, contractors, engineers, and anyone interested in modern plumbing solutions, this guide provides valuable insights into why trenchless pipe repair is becoming the preferred choice for pipe rehabilitation. Stay informed about the latest advancements and best practices in the field.
NO1 Uk best vashikaran specialist in delhi vashikaran baba near me online vas...Amil Baba Dawood bangali
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Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Water scarcity is the lack of fresh water resources to meet the standard water demand. There are two type of water scarcity. One is physical. The other is economic water scarcity.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
state space representation,State Space Model Controllability and Observability Derive Transfer Function from State Space Equation Time Response and State Transition Matrix Eigen Value
1. State Space Representation
1
State Variables of a Dynamical System
State Variable Equation
Why State space approach
Block DiagramRepresentation Of State Space Model
Controllabilityand Observability
Derive Transfer Function from State Space Equation
Time Response and State Transition Matrix
Eigen Value
2. Introduction
2
The state-space description provide the dynamics as a set of coupled
first-order differential equations in a set of internal variables known as
state variables, together with a set of algebraic equations that combine
the state variables into physical output variables.
3. Definition of System State
(𝒙𝟏, 𝒙𝟐, … … , 𝒙𝒏) (called State Variables or State Vector) such that knowledge of
these variables at 𝑡 = 𝑡0, together with knowledge of the input for 𝑡 ≥ 𝑡0 ,
completely determines the behavior of the system for any time t to t0 .
The number of state variables to completely define the dynamics of the system is
equal to the number of integrators involved in the system (System Order).
Assume that a multiple-input, multiple-output system involves n integrators (State
Variables).
Assume also that there are r inputs u1(t), u2(t),……. ur(t) and p outputs y1(t),
y2(t), …….. yp(t). 4
Inner state variables
x1,x2,xn
State: The state of a dynamic system is the smallest set of variables
u1(t)
u2(t)
u (t)
r
y1(t)
y2(t)
yp(t)
5. State-Space Equations (Model)
State equation:
Output equation:
6
x
(t) Ax(t) Bu(t)
y(t) C x(t) Du(t)
Dynamic equations
1
n1
n
x (t)
x (t)
x(t) x2 (t)
1
r1
r
u (t)
u (t)
u(t) u2 (t)
1
p1
p
y (t)
y (t)
y(t) y2 (t)
State Vector
State variable
Input Vector Output Vector
A n n
B nr
C pn
1
n1
n
x (0)
x (0)
x(0) x2 (0)
D pr
6. Block Diagram Representation Of State Space Model
C
A
D
B
1
s
6
+
+
+
+
u(t) y(t)
x(t)
x
(t)
x
(t) Ax(t) Bu(t)
y(t) C x(t) Du(t)
7. sXs AX s BU s
Ys CX s DUs
Then, the transfer function is
X s sI A1
BU s
The state space model
x
Ax Bu
y Cx Du
by Laplace transform
Ys
CsI A1
B D
Us
Us
Ts
Ys CsI A1
B D
State Space model to Transfer Function
8. Controllability
Plant:
x
Ax Bu, x Rn
y Cx Du
Definition of Controllability
A system is said to be (state) controllable at time t0 , if
8
0, 1
t ]
there exists a finite t1 t0 such for any x(t0 ) and any x1 ,
there exist an input u[t that will transfer the state 0
x(t )
x1
to the state at time t1 , otherwise the system is said to
be uncontrollable at time t0 .
9. Controllability Matrix
Consider a single-input system (u ∈ R):
The Controllability Matrix is defined as
We say that the above system is controllable if its controllability matrix
𝐶(𝐴, 𝐵) is invertible.
Condition for controllability(system is controllable if’f)
det(C ) 0 rank(C) n, full rank
9
10. Observability
Plant:
x
Ax Bu, x Rn
y Cx Du
Definition of Observability
and the
10
unobservable at
A system is said to be (completely state) observable at
time t0 , if there exists a finite t1 t0 such that for any x(t0 )
0, 1
t ]
output over the time interval [t0 ,t1] suffices to
determine the state x0 , otherwise the system is said to be
t0 .
at time t0 , the knowledge of the input u[t
0, 1
y[t t ]
12. Controllability and Observability
Theorem I
x
c (t) Ac xc (t) Bc u(t)
Controllable canonical form Controllable
Theorem II x
o (t) Ao xo (t) Bo u(t)
y(t) Co xo (t)
Observable canonical form Observable
A system in Controller Canonical Form (CCF) is always controllable!!
12
A system in Observable Canonical Form (OCF) is always controllable!!
13. Controllable Canonical Form
We consider the following state-space representation, being called a
controllable canonical form, as
Note that the controllable canonical form is important in discussing the
pole-placement approach to the control system design.
13
14. Observable Canonical Form
We consider the following state-space representation, being called an
observable canonical form, as
14
15. Diagonal Canonical Form
Diagonal Canonical Form greatly simplifies the task of computing the
analytical solution to the response to initial conditions.
15
16. State transition matrix
State transition matrix
x
(t) Ax(t)
sX(s) x(0) AX (s)
X (s) (sI A)1
x(0)
x(t) L1
[(sI A)1
]x(0)
eAt
x(0)
(t)eAt
L1
[(sI A)1
]
0
0
0
0
0
At
x(t0 ) (t t0 )x(t0 )
x(t ) e
x(t) e e
x(0) e x(t )
x(t0 ) e 0
x(0)
A(tt )
At At
At
16
17. Eigen values
The eigenvalues of an nxn matrix A are the roots of the characteristic equation.
Consider, for example, the following matrix A:
19. General State Representation
1. Select a particular subset of all possible system variables, and call
state variables.
2. For nth-order, write n simultaneous, first-order differential equations
in terms of the state variables (state equations).
3. If we know the initial condition of all of the state variables at 𝑡0 as
well as the system input for 𝑡 ≥ 𝑡0, we can solve the equations
20. State-Space Representation of nth-Order Systems of Linear
Differential Equations
Consider the following nth-order system:
(𝒏) (𝒏−𝟏)
𝒚 + 𝒂𝟏 𝒚 + … + 𝒂𝒏−𝟏𝒚 + 𝒂𝒏 𝒚 = 𝒖
where y is the system output and u is the input of the System.
The system is nth-order, then it has n-integrators (State Variables)
Let us define n-State variables
20
21. State-Space Representation of nth-Order Systems of Linear
Differential Equations (Cont.)
Then the last Equation can be written as
21
22. State-Space Representation of nth-Order Systems of Linear
Differential Equations (Cont.)
Then, the stat-space state equation is
where
22
23. State Space Model (Example)
Find the state space model for a system that described by the following
differential equation
c
9c
26c
24c 24r
Solution:
The system is 3rd order, then it has three states as follows
x1 c x1 x2
x2 x3
x
3 24x1 26x2 9x3 24r
x2 c
x3 c
The output equation is
y c x1
differentiation
27. System Poles from State Space model
poles and check the stability of the following state space Example find the
System model
Solution:
Since
To find the poles
Then the poles are {-1, -2 }, the system is stable
5
0 2
x
1 3 x 0u y
1 0x
s 2
sI A s(s 3) 2 0
1 s 3
1
2
s 3
sI A
s
28. controllability and Obervability (Example)
28
Hence the system is both controllable and observable.
1,
0
1
0 1
C 0 1
0, B
A
Plant:
x
Ax Bu, x Rn
y Cx Du
1
Obervability Matrix
0
1
1
CA 1 0
N
C
0
Controllability Matrix V B AB
0
rank(V ) rank(N) 2
29. Example
29
c
c
c
y 2 1x
1
0
u
1
x
2 3
x
0
Controllable canonical form
2 1
1
3
1
1
CA
V
C
2
U B AB
0 rank[U] 2 n
rank[V] 1 n
o
o
o
y 0 1x
2
x 1u
1 3
0 2
x
Observable canonical form
3
1
1 1
2 2
CA 1
V
C
0
U B AB
rank[U] 1 n
rank[V] 2 n
(s 1)(s 2)
T(s)
s 2