System Modelling with MATLAB , Basic [NIse, Norman S. 2000]

SYSTEM MODELING

• Use for analysis and design of feedback control system
• Consist of two approaches:

1. Classical or frequency-domain method which converting a differential equation system to transfer function using Laplace transform or z transform
o Simplify the representation of individual subsystem and modeling interconnection subsystems
o Disadvantage: limited applicability only to linear, Time-invariant system
o Advantage: rapidly provide stability and transient response information therefore can immediately see the effect of varying system parameters until an acceptable design is met
o Few quick calculation or graphic representation of data rapidly yields the physical interpretation

2. State-space method or modern or time-domain approach
o Increase in space exploration and involves modeling using linear, time-invariant differential equation.
o MIMO (multiple output multiple output) system is compactly represented similar to single input-output system
o Can handle system with non-zero initial condition
o This time-domain approaches can be used to represent system via digital computer for digital simulation where system response can be obtained for changes in system parameters.
o Numerous state-space software packages offered for PC
o Disadvantage: Designer engaged in several calculations before the physical representation of the model is apparent.

• LINEAR
• TIME VARIANT
  

• LAPLACE TRANSFORM
• Use table and theorem for transformation from time domain to frequency domain
• Partial fraction expansion can be applied to simplify a complicated function by braking it down to a sum of simpler term
• F(S)=N(S)/D(S)
• The order of N(S) must be less than D(S) if not N(S) must be divided by D(S)
• Case 1: Roots of the Denominator of F(S) Are Real and Distinct
• Case 2: Roots of the Denominator of F(S) Are Real and Repeat
• Case 3: Roots of the Denominator of F(S) Are Complex or Imaginary

• TRANSFER FUNCTION
• H(S)=C(S)/Y(S) r = reference input, c = control output
• Differential equation (time domain) transform using Laplace (frequency domain) therefore allow separation of input, system, and output.
• Algebraically relates a system’s output to its input.
• Allow separation of the input, system, and output into three separate and distinct parts.
• Algebraically combine mathematical representations of subsystems to yield a total system representation.
  

• LINERIZATION
• Obtained linear approximation for nonlinear system in order to obtained transfer function

• STATE SPACE

MATLAB CODE TRANSFER FUNCTION STATE SPACE