State Space Models

The same idea as State Variable Models (literally the same thing).

Example

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Here, we want to derive equations of motion using Newtons 2nd Law

• $M{y}^{\prime \prime }=u-D{y}^{\prime }$
• We can put this in a standard form which requires only 1st order derivatives of our variables
• This standard form has many advantages for stability analysis and control design called state space models or representations To re-write this 2nd order ODE as a first oder ODE, we need 2 states (State Variable Models).
• $x=[x_1,x_L]$
• $x_1:=y$
• $x_L=y^{\prime }$
• $x_1^{\prime }=y^{\prime }=x_2$
• $x_L^{\prime }=y^{\prime \prime }=\frac{1}{M}(u-D{y}^{\prime })=\frac{1}{m}(u-D{x}_2)=-\frac{D}{M}{x}_2+\frac{1}{M}u$
• More compactly…
• $x^{\prime }=\left[\begin{array}{c}{x}_1^{\prime }\\ x_2^{\prime }\end{array}\right]=\left[\begin{array}{ccc}0& & 1\\ 0& & -\frac{D}{M}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+\left[\begin{array}{c}0\\ \frac{1}{M}\end{array}\right]$ ⇒ $x^{\prime }=Ax+Bu$
• Also, $y=x_1$
  • $y=[1,0]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]+0u$
    • Where here the 1, 0 matrix is C, and the 0 is D Overall, we have:

$x^{\prime }=Ax+Bu$ and $y=Cx+Du$ which is the standard state space representation for LTI systems where A, B, C, D are constant matrices

Uniqueness

These representations are not unique!

Summary

$dimension(=\text{number of states})\text{of a state space model is not unique either!}$ Note that you can infinitely add redundant states making a new state space model that is the same as the last.

Definition

An LTI state space model of a system is a model of the form:

• $x^{\prime }=Ax+Bu$
• $y=Cx+Du$ Such that for any initial condition $x(t=0)=x_o$ and any input signal $u(t)$, the system output is equal to the output $y(t)$ of our standard state space model form.

To Frequency Domain

$x^{\prime }=Ax+Bu$
$y=Cx+Du$

Using $sx=sIX$ becomes…
$sX=AX+BU$ $Y=CX+DU$

and…
$X=(sI-A{)}^{-1}BU$ $Y=T_{uy}(s)U(s)$ where $T_{uy}=C(sI-A{)}^{-1}B+D$

Definition

Let $T_{uy}$ be real, rational, and proper.
Then a state-space realization of $T_{uy}$ is an LTI state space model such that $T_{uy}=C(sI-A{)}^{-1}B+D$ or such that you can recovery $T_{uy}$

Note that state space realizations are not unique.

Here, I is the identity matrix.

Related

• Finding State Space Realizations
• Nonlinear State Space Models