Linear System Model

A) Linear system with one state variable, one input and one output

Any Linear system can be described through linear model as below:

• Specify the state
• ex: position (x)
• Specify the inputs
• ex: velocity (v)
• Specify the output
• ex: position (x)

You will have your system described as follows:

• Differential equation

• Output equation

y = x

B) Linear system with multiple state variable, one input and one output

Suppose that you have a linear system with:

• State: x, y, PHI
• i.e: x1 = x, x2=y, x3=Φ
• Inputs: nu, PHI
• i.e: u1 = nu, u2 = Φ
• Output: x,y,PHI
• i.e: y1 = x, y2=y, y3=Φ

You will have your system described as follows:

• Differential equations

• Output equations

The above equations can be written using Matrices as follows:

• Differential equations

• Output equations

in our example, C11, C22, C33 are equal to one, others are zeros

C) Nonlinear systems (Linearization)

If we have systems that are described by nonlinear differential equations, where you can find terms like:

• x1*x2
• x1*u1
• cos(x3)
• ...etc

We can linearize them around a certain point of operation (x0,u0)

• Equations will be in the form:

We can linearize this around x0, u0 (specific state and specific input)

Note, the above equations assumes that x0, u0 satisfies: f(x0,u0) = 0, h(x0) = 0