State Variable Models

State Variable Models

State variable models are made of 1st order ODEs instead of higher order equations.

They allow linear algebra to be used

S.V models are formed by defining higher order derivates as a cascade of 1st order derivatives.

Let’s define some variables…
$x_1\left(t\right)=x\left(t\right)$
$x_2\left(t\right)=x_1^{\prime }\left(t\right)=x^{\prime }\left(t\right)$ $x_3\left(t\right)=x_2^{\prime }\left(t\right)=x^{{}^{\prime \prime }}\left(t\right)$ And this trend continues, as we can see, higher order derivatives can be represented as a first order derivative of another variable…

A state variable model is basically an equation of motion using these state variable instead of regular variables

See the OneNote for examples

Your state variable model will look something like this:
$x_1^{\prime }=x_2\left(t\right)$ $x_2^{\prime }=x_3\left(t\right)$
$x_3^{\prime }\left(t\right)=\frac{f\left(t\right)}{a_3}-\frac{a_2}{a_3}{x}_3\left(t\right)-\frac{a_1}{a_3}{x}_2\left(t\right)-\frac{a_0}{a_3}{x}_1\left(t\right)$
Where you put this into a matrix modelling all three equations as described below.

Procedure

After creating a state variable model, you put it in matrix form where your state variable first derivatives are in a vector on one side, and your coefficients multiplied by the state variable vector is on the other side.

The Standard State Variable Model
$\left\{x^{\prime }\right\}=\left[A\right]\left\{x\right\}+\left[B\right]\left\{u\right\}$
Where:

• A: State matrix
• x: State vector
• B: input (control) matrix
• u: Input vector

The output form is basically the same except with output state matrices and output vectors…

You can also put an equation of motion into the LaPlace domain and then represent that as a state variable model and then represent it in matrix form.

Solution Method

• Isolate the “1” term in denominator
• Find the output of the integrator and call it a state (where in LaPlace the integrator is 1/s and the output term it is attached to)

State-Variable Model of Circuits

1. Find storage elements (L & C)
2. Consider stored energy as $\frac{C{v}^2}{2}$ and $\frac{L{i}^2}{2}$

Tips for modelling circuit elements using state variables

The goal here is to create an ODE that has derivatives of the output variable(s) with no derivatives of the input variable(s)

For C and L, we go with the expression that works with the output (for example, if V is out output we would use the v expression for C and L)

The representation of C and L come from Impedance

We use KVL and KCL to form these expressions and models

We can also take our time derivatives to the LaPlace domain to solve.

The Impedance Method

Basically using ohms law and equivalent impedance to create transfer functions.

1. Find $Z_{eq}$
2. Define $\frac{V_s\left(s\right)}{I\left(s\right)}=Z_{eq}$
3. Define output with respect to a known term
4. Perform substitution and solve for the required transfer function

Force to Voltage Analogy

For a mass, spring damper system:
$f=m\frac{dv}{dt}+cv+k{\int }_{}^{}vdt$
For an LRC circuit:
$V_s=L\frac{di}{dt}+Ri+\frac{1}{C}{\int }_{}^{}idt$
You can view the similarities by comparing constants and independent variables

We can also create transfer functions of ${\frac{V_O}{V_1}}_{}$ in the same manner as we do in mechanical systems and signals.