K10880 deepaknagar control slide share
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This document contains a major assignment on controllability and observability submitted by Deepak Nagar. It includes: 1) Definitions of controllability and observability of linear systems and conditions for a system to be completely controllable and observable. 2) Proofs showing how to determine controllability and observability by calculating the rank of controllability and observability matrices. 3) Examples demonstrating how to put systems into controllable and observable canonical forms and the relationship between these forms and the rank of the matrices.
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ELEG 421 Control Systems Transient and Steady State .docx
ELEG 421
Control Systems
Transient and Steady State
Response Analyses
Dr. Ashraf A. Zaher
American University of Kuwait
College of Arts and Science
Department of Electrical and Computer Engineering
Layout
2
Objectives
This chapter introduces the analysis of the time response of different
control systems under different scenarios. Only first and second order
systems will be considered in details using analytical and numerical
methods. Extension to higher order systems will be developed. Both
transient and steady state responses will be evaluated. Stability analysis
will be analyzed for different kinds of feedback, while investigating the
effect of both proportional and derivative control actions on the
performance of the closed-loop system. Finally systems types and
steady state errors will be calculated for unity feedback.
Outcomes
By the end of this chapter, students will be able to:
evaluate both transient/steady state responses for control systems,
analyze the stability of closed-loop LTI systems,
investigate the effect of P and I control actions on performance, and
understand dominant dynamics of higher order systems.
Dr. Ashraf Zaher
Introduction
3
Test signals
Transient response
Steady state response
Analytical techniques, and
Numerical (simulation) techniques.
Stability (definition and analysis methods),
Relative stability, and
Effect of P/I control actions on stability and performance.
Summary of the used systems:
First order systems,
Second order systems, and
Higher order systems.
Dr. Ashraf Zaher
Test Signals
4 Dr. Ashraf Zaher
Impulse function:
Used to simulate shock inputs,
Laplace transform: 1.
Step function:
Used to simulate sudden disturbances,
Laplace transform: 1/s.
Ramp function:
Used to simulate gradually changing inputs,
Laplace transform: 1/s2.
Sinusoidal function(s):
Used to test response to a certain frequency,
Laplace transform: s/(s2+ω2) for cos(ωt) and ω/(s2+ω2) for sin(ωt).
White noise function:
Used to simulate random noise,
It is a stochastic signal that is easier to deal with in the time domain.
Total response:
C(s) = R(s)*TF(s) = Ctr(s) + Css(s) → c(t) = ctr(t) + css(t)
Fundamentals
5 Dr. Ashraf Zaher
Definitions:
Zeros (Z) of the TF
Poles (P) of the TF
Transient Response (Natural)
Steady State Response (Forced)
Total Response
Limits:
Initial values
Final values
Systems (?Zs):
First order (one P)
Second order (two Ps)
Higher order!
More:
Stability and relative stability
Steady state errors (unity feedback)
First Order Systems
6 Dr. Ashraf Zaher
TF:
T: time constant
Unit Step Response:
1
1
)(
)(
+
=
TssR
sC
)/1(
11
1
1
1
11
)(
TssTs
T
sTss
sC
+
−=
+
−=
+
=
Ttetc /1)( −−=
632.01)( 1 =−== −eTtc
T
e
Tdt
tdc Tt
t
11)( /
0
== −
=
01)0( 0 =−== etc
11)( =−=∞= −∞etc
First Order Systems.
14599404.ppt
This document provides an introduction to state space modeling and analysis in modern control systems. It defines key concepts such as state, state variables, state vector, and state space. The document describes how to represent dynamic systems using state space equations involving state variables, input variables, output variables, and state vectors. It provides examples of modeling a mechanical system and DC motor in state space form. It also discusses the concept of state controllability and how to determine if a system is completely controllable using the controllability matrix.
Control assignment#4
This document contains 8 problems related to systems engineering. The problems cover various topics like determining states and outputs of linear systems given state space representations, controllability/observability analysis, designing state feedback controllers and observers, and computing step responses of closed loop systems.
Robust Adaptive Controller for Uncertain Nonlinear Systems
The document presents a comparison of adaptive backstepping control and adaptive backstepping sliding mode control for uncertain nonlinear systems transformed into semi-strict feedback form. It describes the design of an adaptive backstepping controller using a recursive algorithm to stabilize the system and ensure output tracking. An adaptive backstepping sliding mode controller is also designed combining the backstepping technique with sliding mode control. Simulation results show that the adaptive backstepping sliding mode control has better tracking performance and parameter estimation ability, especially when the unknown parameters are time-varying, demonstrating its greater robustness.
Lyapunov stability
1. The document discusses Lypunov stability and different types of stability including asymptotically stable, bounded-input bounded-output stable, and Lyapunov stability.
2. It provides conditions for asymptotic stability including having all eigenvalues of the system in the left half plane and defines an equilibrium state as a state where the system will not move from in the absence of input.
3. Lyapunov's method is introduced for analyzing stability using a Lyapunov function where the derivative must be negative semi-definite to guarantee asymptotic stability.
EC202 SIGNALS & SYSTEMS Module1 QUESTION BANK
This document contains a question bank with 10 questions related to signals and systems for the module EC 202. The questions cover topics like determining if a signal is periodic, plotting signals, identifying if signals are energy or power signals, analyzing properties of systems like time invariance and causality, explaining classifications of signals, properties of systems, types of systems, and decomposing a signal into odd and even components.
Controllability of Linear Dynamical System
The document discusses linear dynamical systems and controllability of linear systems. It defines dynamical systems as mathematical models describing the temporal evolution of a system. Linear dynamical systems are ones where the evaluation functions are linear. Controllability refers to the ability to steer a system from any initial state to any final state using input controls. The document provides the definition of controllability for linear time-variant systems using the controllability Gramian matrix. It also gives the formula for the minimum-norm control input that can steer the system between any two states. An example of checking controllability for a time-invariant linear system is presented.
Msc lecture07
The document discusses various concepts related to stability analysis of control systems including:
1) Bounded-input bounded-output (BIBO) stability, asymptotic stability, and Lyapunov stability.
2) For linear systems, BIBO stability is equivalent to having all poles in the left half plane, while asymptotic stability requires all eigenvalues of the A matrix to have negative real parts.
3) Lyapunov's stability theorem provides a method to analyze stability using a Lyapunov function that is positive definite and whose derivative is negative semi-definite.
state space representation,State Space Model Controllability and Observabilit...
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
Time Domain Specifications
This document discusses time response specifications for second order systems, including delay time, rise time, peak time, and peak overshoot. It provides equations to calculate each specification based on the natural frequency (ωn) and damping ratio (δ) of the system. Rise time is defined as the time to rise from 10-90% of the final value and is calculated as (π - cos^-1δ)/ωn√(1-δ^2). Peak time is the time to reach the first peak and is calculated as π/ωn√(1-δ^2). Peak overshoot is calculated as 100e^(-δπ)/√(1-δ^2)
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K10880 deepaknagar control slide share
- 1. CAREER POINT UNIVERSITY MAJOR ASSIGNMENT ON CONTROLABILITY AND OBSERVABILITY SUBMITTED TO MR. SOMESH CHATURVEDY SUBMITTED BY Deepak Nagar KID-K10880 B.Tech. (M.E.) 6th Semester
- 2. Motivation1 [ ] = + − − = 2 1 2 1 2 1 01 )( 0 1 10 12 x x y tu x x x x 1− s 1− s 1 1− 2− u y1x2x s x )0(2 s x )0(1 1 1x2x 1 controllable uncontrollable
- 3. Motivation2 [ ] = + − − = 2 1 2 1 2 1 01 )( 1 3 10 02 x x y tu x x x x 1− s 1− s 1 1− 2− u y1x2x s x )0(2 s x )0(1 1 1x2x 3 observable unobservable
- 4. Definition A linear system is said to be completely controllable if, for all initial times and all initial states , there exists some input function (or sequence for discrete systems) that drives the state vector to any final state at some finite time . 0t )( 0tx )( 1tx 10 tt < A linear system is said to be completely observable if, for all initial times , the state vector can be determined from the output function (or sequence) , defined over a finite time . Definition 0t )( 0tx )( 1ty 10 tt <
- 5. Proof of controllability matrix [ ] =− ++++=− +++++= ++=++= += += −+ −+ − + −+−++ −− + −+−++ −− + +++ +++ + )1( )2( 1 )1()2(1 21 )1()2(1 21 1 2 12 112 1 )( nk nk k n k n nk nknkk n k n k n nk nknkk n k n k n nk kkkkkkk kkk kkk u u u BABBAxAx BuABuBuABuAxAx BuABuBuABuAxAx BuABuxABuBuAxAx BuAxx BuAxx Initial condition
- 6. Proof of observability matrix ( ) [ ])1()2()3(11 1 )1()2(1 321 1 111 111 1 )(),2(),1( )( )2()( )1( −+−+−+++ − −+−++ −−− −+ +++ +++ + −−−−−= ⇒ +++++= ++=++= += += += nknknkkkkkk k n nknkk n k n k n nk kkkkkkk kkk kkk kkk DuCBuCABuDuCBuyDuy x CA CA C n nDuCBuBuCABuCAxCAy DuCBuCAxDuBuAxCy DuCxy DuCxy BuAxx Inputs & outputs
- 7. )()()( )()()( tDutCxty tButAxtx += += [ ] = = − − 1 2 12 n n CA CA CA C V BABAABBU Controllability matrix Observability matrix Then: (1) controllable (2) observable nVrank nUrank = = )( )(
- 8. [ ] = + − − = 2 1 2 1 2 1 01 )( 0 1 10 12 x x y tu x x x x Example 1 2][ 12 01 1][ 00 21 ][ = − = = = − == Vrank CA C V UrankABBU uncontrollable observable The rank of a matrix is defined by the number of linearly independent rows and/or the number of linearly independent columns
- 9. Example 2 [ ] = + − − = 2 1 2 1 2 1 01 )( 1 3 10 02 x x y tu x x x x 1][ 02 01 2][ 11 63 ][ = − = = = − − == Vrank CA C V UrankABBU controllable unobservable
- 10. Theorem III )()()( tuBtxAtx cccc += Controllable canonical form Controllable Theorem IV )()( )()()( txCty tuBtxAtx oo oooo = += Observable canonical form Observable
- 11. example [ ] c cc xy uxx 12 1 0 32 10 = + −− = Controllable canonical form [ ] −− = = − == 12 12 31 10 CA C V ABBU nVrank nUrank ≠= == 1][ 2][ [ ] o oo xy uxx 10 1 2 31 20 = + − − = Observable canonical form [ ] − = = − − == 31 10 11 22 CA C V ABBU nVrank nUrank == ≠= 2][ 1][ )2)(1( 2 )( ++ + = ss s sT
- 12. Theorem V )()()( )()()( tDutCxty tButJxtx += += Jordan form [ ]321 3 2 1 3 2 1 CCCC B B B B J J J J = = = Jordan block Least row has no zero row First column has no zero column
- 13. Example [ ]xccy ub b xx 3 1100 020 012 1211 12 11 = + = If 012 =b uncontrollable If 011 =c unobservable
- 15. { } { } { } { } .... .... 21131211 21131211 ILCILCCC ILbILbbb + + controllable observable In the previous example { } { } { } { } .... .... 21131211 21131211 DLCILCCC ILbILbbb + + controllable unobservable