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Modeling in the Time Domain - Coggle Diagram
Modeling in the Time Domain
Introduction
Two approaches for analysis and design of feedback control systems
Classical (frequency domain) technique
Algebraic
Limited applicability to linear, time-invariant systems
Rapidly provide stability and transient response information
State-space approach (time-domain)
Can be used to represent nonlinear, time-varying, and MIMO systems
Can handle nonzero initial conditions
Approach
Select state variables: subset of all possible system variables
Minimum number of state variables equal to order of diff eq
Write state equations: n simultaneous, 1st order diff eq in terms of state variables
If IC of all state variables at t_0 and system input for t >= t_0 solve simultaneous diff ew for all state variables for t >= t_0
Output equation: algebraically combine (linear combination) state variables with system input and find all other system variables for t >= t_0
State-space representation: state equations and output equations are viable representation of system
The General State-Space Representation
Linear combination
Linear independence: none of the variables can be written as linear combination of others
System variable: any variable that responds to input or IC
State variables: smallest set of linearly independent system variables such that value of members of set at time t_0 along with known forcing function completely determine value of all system variables for all t >= t_0
State vector: elements are state variables
State space: n-dimensional space whose axes are state variables
State equations
Output equation
Applying the State-Space Representation
Select state vector
Components must be linearly independent
Must select minimum number of state variables
Equal to order of diff eq and can be found by counting energy-storage elements in system
Converting a Function to State Space
Phase variables: each subsequent state variable is defined to be the derivative of the previous state variable
Phase variable choice: choose the output, y(t), and its (n − 1)
derivatives as the state variables
Convert transfer function to diff eq by cross-multiplying and taking inverse Laplace transform, assume 0 IC
Represent in phase-variable form
