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Chapter 3: Modeling in the Time Domain - Coggle Diagram
Chapter 3: Modeling in the Time Domain
3.1 Introduction
Two Approaches available to model a Control System
Classical or Frequency Domain
Converts differential equations to transfer functions
Primary disadvantage is limited applicability, only can be applied to linear, time-invaraiant system
Major advantage they rapidly provide stability and transient response information
Modern or Time Domain
Also reffered to as state space approach
Unified method for modeling, analyzing, and designing a wide range of systems
Can be used on non-linear systems
Can be used on multiple inputs and multiple outputs
3.2 Some Observations
The examples in the textbook use the following approaches
If we know the intiial condition af all of the state variable at t0 as well as the system input for time greater than 0, we can solve for the simultaneous differential equations for the state variables for time greater than 0
We algebraically combine the state variables with the system's input and find all of the other system variables for time greater than 0, called
output equation
For an nth-order system, we write n simultaneous, first-order differential equations, called
state equations
We consider the state equations and the output equations a variable representation of the system, this is called
state-space representation
Select a particular subset of all possible system variable, call the variable
state variables
3.3 The General State-Space Representation
Represented by equations:
x' = derivative of the state vector with respect to time
y = output vector
x = state vector
u = input or control vector
B= input matrix
C= output matrix
A = system matrix
D= feedforward matrix
3.4 Applying the State-Space Representation
Typically the min number of variables required equals the order of the differential equation describing the system
Number of independent energy storage devices (inductors and capacitor) equals the order of the differential equation for the system
Too few variables and the system is unsolvable
3.5 Converting a Transfer Function to State Space
Convient way is to choose the output and its (n-1) derivatives
This choice is called phase-variable choice
3.6 Converting from State Space to a Transfer Function