State space analysis
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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.
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This document discusses system modeling and properties of linearity and time invariance. It provides examples of modeling a resistor, square-law system, delay operator, and time compressor to illustrate these properties. A model is a set of mathematical equations relating the output and input signals of a physical system. A system is linear if the response to a sum of inputs is the sum of the responses. It is time invariant if its behavior does not change over time. Developing an accurate but simplified model is important for understanding system behavior and designing controllers.
PhDretreat2011
1. The document discusses methods for calculating summary statistics for continuous-time Markov chains (CTMCs), which are used to model processes like DNA sequence evolution.
2. It focuses on calculating the expected number of jumps between states and expected waiting time in a state, conditioned on the start and end points. These statistics are needed to estimate model parameters using maximum likelihood approaches like the EM algorithm.
3. As an application, the document describes using a 61x61 CTMC rate matrix to model codon sequence evolution, with the matrix parameters estimated via the EM algorithm using the summary statistics. Expected jumps between states differentiated by transition type are of particular interest.
State space model
This document introduces linear time-varying (LTV) systems and the computation of the state transition matrix (STM) for LTV systems. It discusses:
1) The definition and properties of the STM for LTV systems.
2) Conditions under which the system matrix A(t) commutes with the integral of A(t), which is required to compute the STM.
3) Examples of computing the STM for different time-varying system matrices.
4) How to obtain the overall state solution for an LTV system given its STM.
5) An introduction to discretizing continuous-time systems.
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
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State space analysis
- 1. State Space Analysis Introduction and Terminologies
- 2. Basic Laws of Physics Set of Differential Equations State Space Method Transfer Function Method
- 3. Electrical System ● E(t) and I(t) are called characterising variables ● Energy stored in capacitor =1/2(CV^2) ● Energy stored in inductor= ½(LI^2) I e Ei
- 4. ● e(t) and i(t) can be found for t>=0 ● If initial energy state, e(0) and i(0) as well as, Ei(t) for t>=0 are known. ● So they can be defined as state variable. ● State variable:- They are set of characterising variable which gives you total information about at any time instant t>=0, provided initial state and external input are known to you.
- 5. ● From Basic laws of physics, Ei = Ri + e + L di/dt Also i=C de/dt. ● Mathematical Model, de(t)/dt= 1/C i(t) di/dt= R/L i(t) + 1/L e(t) + 1/L Ei(t) Ei i e
- 6. ● According to standard nomenclature, Assume x1 = e(t) , x2=i(t) and r=Ei(t) ● State equations, dx1/dt= 1/C x2 dx2/dt = R/L x2 + 1/L x1 +1/L r ● If output we require is voltage across an inductor, y(t) = -Ri - e + Ei = -R x2 - x1 + r State Space Model y(t)
- 7. Mechanical System Assume , x1=x(t) and x2=v(t) ● consider that we want to measure the position of the body. y=x1
- 8. Standard Format Where , ● x(t) : "state vector" ● y(t) : "output vector" ● u(t) : "input (or control) vector" ● A : "state (or system) matrix" ● B : “input matrix" ● C : "output matrix" ● D :"feedthrough (or feedforward) matrix"
- 9. ● State variable defined are not unique. They can be defined by almost infinite ways. ● We can define state variable according to our convenience. Eg. Charge stored in capacitor. ● State variable formulation will give you unique output for given input. Points to be considered
- 10. Advantages ● Complex techniques. ● Many computations are required. ● State model is not the unique property of the system, because the order of the state variables is not important due to which state model matrices get changed. Disadvantages ● It can be applied to nonlinear system. ● It can be applied to time invariant systems. ● It can be applied to multiple input multiple output systems. ● Its gives idea about the internal state of the system. ● Can be used in time domain.
- 11. 1.