State-Variable Models

State-variable models are made of first-order differential equations, allowing linear algebra to be used to solve problems. They are formed by defining higher-order derivatives as a cascade of first-order derivatives.

Generally, we have various time derivatives of some function $x$:

$$x(t),\dot{x}(t),\ddot{x}(t),\text{dddot}x(t)$$

We then use:

$$\begin{array}{rl}{x}_1(t)& =x(t)\\ x_2(t)& ={\dot{x}}_1(t)=\dot{x}(t)\\ x_3(t)& ={\dot{x}}_2(t)=\ddot{x}(t)\\ x_4(t)& ={\dot{x}}_3(t)=\text{dddot}x(t)\end{array}$$

Examples

Mass-Spring-Damper

An example is the typical mass-spring-damper system:

$$\begin{array}{rl}& \\ m\ddot{x}+c\dot{x}+kx& =f\\ \ddot{x}+\frac{c}{m}\dot{x}+\frac{k}{m}x& =\frac{f}{m}\end{array}$$

We define:

$$\begin{array}{rl}{x}_1& =x\\ x_2& ={\dot{x}}_1=\dot{x}\\ x_3& =\dot{x_2}=\ddot{x}\end{array}$$

Then, we can re-write the original equation as:

$$\begin{array}{rl}{\dot{x}}_2+\frac{c}{m}{x}_2+\frac{k}{m}{x}_1& =\frac{f}{m}\\ {\dot{x}}_2& =\frac{f}{m}-\frac{c}{m}{x}_2-\frac{k}{m}{x}_1\end{array}$$

Thus, we’ve reduced our original 2nd-order equation to two first-order equations:

$$\{\begin{array}{l}{\dot{x}}_2=\frac{f}{m}-\frac{c}{m}{x}_2-\frac{k}{m}{x}_1\\ {\dot{x}}_1=x_2\end{array}$$

This is our state-variable model. The variables $x_1$ and $x_2$ are the state variables.

3rd Order System

A third order system:

$$a_3\text{dddot}y(t)+a_2\ddot{y}(t)+a_1\dot{y}(t)+a_{0}y(t)=f(t)$$

Re-arrange so that highest order variable is isolated:

$$\text{dddot}y(t)=\frac{1}{a_3}[f(t)-a_2\ddot{y}(t)-a_1\dot{y}(t)-a_{0}y(t)]$$

Since this is 3rd order, we expect three 1st-order ODEs to describe the system.

We define

$$\begin{array}{rl}{x}_1(t)& =y(t)\\ x_2(t)& ={\dot{x}}_1(t)=\dot{y}(t)\\ x_3(t)& ={\dot{x}}_2(t)=\ddot{y}(t)\\ {\dot{x}}_3(t)& =\text{dddot}y(t)\end{array}$$

Now, our state-variable model is::

$$\begin{array}{rl}{\dot{x}}_1(t)& =x_2(t)\\ {\dot{x}}_2(t)& =x_3(t)\\ {\dot{x}}_3(t)& =\frac{f(t)}{a_3}-\frac{a_2}{a_3}{x}_3(t)-\frac{a_1}{a_3}{x}_2(t)-\frac{a_0}{a_3}{x}_1(t)\end{array}$$

In vector-matrix form, we can write:

$$\left[\begin{array}{c}{\dot{x}}_1\\ {\dot{x}}_2\\ {\dot{x}}_3\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ -\frac{a_0}{a_3}& -\frac{a_1}{a_3}& -\frac{a_2}{a_3}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ \frac{1}{a_3}\end{array}\right]f(t)$$

The output $y(t)=x_1(t)$ can also be described in matrix form:

$$y(t)=\left[\begin{array}{ccc}1& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]+\left[\begin{array}{c}0\end{array}\right]f(t)$$

Standard State-Variable Model

The standard form of a state variable model is given by the state equation and the output equation.

State equation:

$$\left\{\begin{array}{c}\dot{x}\end{array}\right\}=\left[\begin{array}{c}A\end{array}\right]\left\{\begin{array}{c}x\end{array}\right\}+\left[\begin{array}{c}B\end{array}\right]\left\{\begin{array}{c}u\end{array}\right\}$$

where:

• $\{\dot{x}\}$ is the derivative of the state vector, $n\times 1$
• $[A]$ is the system/state matrix, $n\times n$
• $\{x\}$ is the state vector, $n\times 1$
• $[B]$ is the input/control matrix, $n\times m$
• $\{u\}$ is the input vector, $m\times 1$

Essentially, $x$ and $u$ are column vectors containing the state variables and the inputs, if any.

Output Equation

Output equation:

$$\left\{\begin{array}{c}y\end{array}\right\}=\left[\begin{array}{c}C\end{array}\right]\left\{\begin{array}{c}x\end{array}\right\}+\left[\begin{array}{c}D\end{array}\right]\left\{\begin{array}{c}u\end{array}\right\}$$

where:

• $\{y\}$ is the output vector, $p\times 1$
• $[C]$ is the state output matrix, $p\times n$
• $\{x\}$ is the state vector, $n\times 1$
• $[D]$ is the control output, $p\times m$
• $\{u\}$ is the input vector, $m\times 1$

Note that $n\ne p$ in general.