Modeling in the Time Domain

1
Modeling in the Time Domain

• Kuliah Sistem Kendali

2
Objectives

• How to find mathematical model, called a
  state-space representation, for a linear,
  time-invariant system
• How to convert between transfer function and
  state space models
• How to linearize a state space representation

3
Plant
Mathematical Model Differential equation
Linear, time invariant
Frequency Domain Technique
Time Domain Technique
4
Two approaches for analysis and design of control
system

1. Classical Technique or Frequency Domain Technique
2. Modern Technique or Time Domain Technique

5

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 as well as the system input
   for , we can solve the equations

6
RL network
7
1. Select
As state variables
(1)
2. Write loop equation
3. Solve the equation using laplace transform
(1) Not unique
Unit step
Assumption
4. We can solve all other network variables
(2)
Output equations
(3)
(1), (2),(3) state-space representation
8
(No Transcript)
9
RLC network
1. State variables
10
2.
Using
(1)
3. Can be solved using Laplace
Transform
4. Other network variables can be obtained
(2)
(1),(2) state-space representation
5.
11
Other variables
Each variables linearly independent
12
In vector-matrix form
(1)
where
(2)
where
13
General State Representation
State equation
output equation
state vector
derivative of the state vector with respect to
time
output vector
input or control vector
system matrix
input matrix
output matrix
feedforward matrix
14
Some definitions

• System variable any variable that responds to
  an input or initial conditions in a system
• State variables the smallest set of linearly
  independent system variables such that the values
  of the members of the set at time t0 along with
  known forcing functions completely determine the
  value of all system variables for all t t0
• State vector a vector whose elements are the
  state variables
• State space the n-dimensional space whose axes
  are the state variables
• State equations a set of first-order
  differential equations with b variables, where
  the n variables to be solved are the state
  variables
• Output equation the algebraic equation that
  expresses the output variables of a system as
  linear combination of the state variables and the
  inputs.

15
Graphic representation of state space and a state
vector
16
Application
Electrical network
17
1. Select state variables
(1)
Express (1) using
2.
(2)
3.
(3)
4.
Output equation
18
Convert a transfer function
choose
diferensiasikan
19
(No Transcript)
20
Example TF to State Space
1.
Inverse Laplace
2.
Select state variables
numerator
denominator
21
(No Transcript)
22
Decomposing a transfer function
23
(No Transcript)
24
Example
25
(No Transcript)
26
State Space to TF
Laplace Transform
27
(No Transcript)
28
(No Transcript)
29
Appendix